🔢 Basic Number Types
1. Natural Numbers
Natural numbers are the numbers you use when you start counting things, like 1 apple, 2 toys, 3 books. They start from 1 and go on forever!
- Start from 1, not 0.
- Used for counting objects.
- Always positive, never negative.
- No fractions or decimals allowed!
- Easy way to remember: If you can count it, it’s a natural number!
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
2. Whole Numbers
Whole numbers are just like natural numbers, but they also include 0. Think of them as natural numbers with a zero added at the start!
- Starts from 0.
- Never includes negative numbers.
- Used for counting items, even if there are none.
- No decimals or fractions.
- If you can count objects, including "nothing," it's a whole number.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
| 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
| 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |
3. Integers
Integers are all the whole numbers and their negatives, including zero. So, you have numbers like -2, -1, 0, 1, 2, and so on! They can be positive or negative, but no fractions or decimals.
- Integers include both positive and negative numbers, plus zero.
- Don’t include fractions or decimals, just whole numbers.
- If you can count forward and backward, you’re using integers!
- They’re great for representing things that can go up or down, like temperature or balance.
| -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |
| 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
🔢 Prime and Related
1. Prime Numbers
Prime numbers are numbers that are only divisible by 1 and themselves. For example, 2, 3, 5, 7, and 11 are prime numbers. They can’t be divided evenly by any other number.
- Prime numbers are only divisible by 1 and themselves.
- The smallest prime number is 2, which is the only even prime number!
- Every number greater than 1 is either prime or can be made by multiplying prime numbers.
- To check if a number is prime, try dividing it by smaller primes like 2, 3, 5, etc.
| 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 |
| 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 |
| 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 |
| 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 |
| 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 |
2. Composite Numbers
Composite numbers are numbers that are not prime. They have more than two factors. For example, 4, 6, 8, 9, and 10 are composite numbers because they can be divided by numbers other than 1 and themselves.
- Composite numbers have factors other than 1 and themselves.
- Any number greater than 1 that is not prime is composite.
- The smallest composite number is 4.
- If a number can be divided by smaller numbers evenly, it is composite.
| 4 | 6 | 8 | 9 | 10 | 12 | 14 | 15 | 16 | 18 |
| 20 | 21 | 22 | 24 | 25 | 26 | 27 | 28 | 30 | 32 |
| 33 | 34 | 35 | 36 | 38 | 39 | 40 | 42 | 44 | 45 |
| 46 | 48 | 49 | 50 | 51 | 52 | 54 | 55 | 56 | 57 |
| 58 | 60 | 62 | 63 | 64 | 65 | 66 | 68 | 69 | 70 |
3. Co-prime Numbers
Co-prime numbers are two numbers that don’t share any common factor (except the number 1). For example, 8 and 15 are co-prime because the only number that divides both of them is 1!
- If two numbers don’t have any same factors (other than 1), they are co-prime!
- One of them can be a prime number, but both don’t have to be.
- Consecutive numbers (like 4 and 5) are always co-prime!
- Use the “greatest common factor” trick—if GCF is 1, they are co-prime.
- Even numbers can’t be co-prime with each other (since they both share 2).
| (2,3) | (3,4) | (4,5) | (5,6) | (6,7) | (7,8) | (8,9) | (9,10) | (10,11) | (11,12) |
| (13,14) | (15,16) | (17,18) | (19,20) | (21,22) | (23,24) | (25,26) | (27,28) | (29,30) | (31,32) |
| (33,34) | (35,36) | (37,38) | (39,40) | (41,42) | (43,44) | (45,46) | (47,48) | (49,50) | (51,52) |
| (53,54) | (55,56) | (57,58) | (59,60) | (61,62) | (63,64) | (65,66) | (67,68) | (69,70) | (71,72) |
| (73,74) | (75,76) | (77,78) | (79,80) | (81,82) | (83,84) | (85,86) | (87,88) | (89,90) | (91,92) |
4. Twin Primes
Twin primes are pairs of prime numbers that have just 2 numbers between them. Like best friends in math who are always close together! For example, 11 and 13.
- Both numbers in the pair must be prime.
- The difference between them is always 2.
- They are like “prime buddies”! If you find one prime, check +2!
- First few are (3, 5), (5, 7), (11, 13), etc.
- You can use a prime list to scan and check for twins.
| (3,5) | (5,7) | (11,13) | (17,19) | (29,31) | (41,43) | (59,61) | (71,73) | (101,103) | (107,109) |
| (137,139) | (149,151) | (179,181) | (191,193) | (197,199) | (227,229) | (239,241) | (269,271) | (281,283) | (311,313) |
| (347,349) | (419,421) | (431,433) | (461,463) | (521,523) | (569,571) | (599,601) | (617,619) | (641,643) | (659,661) |
| (809,811) | (821,823) | (827,829) | (857,859) | (881,883) | (1019,1021) | (1031,1033) | (1049,1051) | (1061,1063) | (1091,1093) |
| (1151,1153) | (1229,1231) | (1277,1279) | (1289,1291) | (1301,1303) | (1319,1321) | (1427,1429) | (1451,1453) | (1481,1483) | (1487,1489) |
5. Mersenne Primes
Mersenne primes are a special type of prime number that are written as 2p − 1. That means you take 2 and raise it to a prime number power, then subtract 1. Like: 2³ - 1 = 7 (a prime!).
- Start with a prime number “p”.
- Plug into the formula 2p − 1.
- Check if the result is prime—if it is, it's a Mersenne prime!
- Not every prime p gives a Mersenne prime.
- They're named after a French monk, Marin Mersenne!
| 3 | 7 | 31 | 127 | 8191 | 131071 | 524287 | 2147483647 | None | None |
| None | None | None | None | None | None | None | None | None | None |
| Only a few Mersenne primes are known because they get very big quickly! | |||||||||
| Use the formula 2p - 1 with prime p values like 2, 3, 5, 7, 13, 17, etc. | |||||||||
| These are special and often used in computing and cryptography! | |||||||||
🌐 Patterned & Interesting Sequences
1. Even Numbers
Even numbers are numbers that can be divided by 2 without any remainder. Examples of even numbers are 2, 4, 6, 8, 10, and so on.
- Even numbers are divisible by 2.
- They always end in 0, 2, 4, 6, or 8.
- Any number that is divisible by 2 is an even number.
- They are always paired with odd numbers.
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 | 40 |
| 42 | 44 | 46 | 48 | 50 | 52 | 54 | 56 | 58 | 60 |
| 62 | 64 | 66 | 68 | 70 | 72 | 74 | 76 | 78 | 80 |
| 82 | 84 | 86 | 88 | 90 | 92 | 94 | 96 | 98 | 100 |
2. Odd Numbers
Odd numbers are numbers that cannot be divided by 2 evenly. They always have a remainder of 1 when divided by 2. Examples of odd numbers are 1, 3, 5, 7, 9, and so on.
- Odd numbers are not divisible by 2.
- They always end in 1, 3, 5, 7, or 9.
- If a number has a remainder of 1 when divided by 2, it’s odd.
- Odd numbers and even numbers alternate with each other.
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |
| 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 |
| 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | 59 |
| 61 | 63 | 65 | 67 | 69 | 71 | 73 | 75 | 77 | 79 |
| 81 | 83 | 85 | 87 | 89 | 91 | 93 | 95 | 97 | 99 |
3. Perfect Numbers
Perfect numbers are special numbers that are equal to the sum of their divisors (excluding the number itself). For example, 6 is perfect because its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.
- Perfect numbers are equal to the sum of their factors (excluding themselves).
- The first few perfect numbers are 6, 28, and 496.
- They are rare and special numbers in math.
| 6 | 28 | 496 | 8128 | 33550336 |
4. Palindromic Numbers
A palindromic number reads the same forward and backward. Just like the word “mom” or “121”! They look like a mirror.
- Look from left to right and right to left — it’s the same!
- All single-digit numbers are palindromes.
- Common in license plates and digital clocks!
- They can be odd or even numbers.
- Check by flipping the digits!
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 11 |
| 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 101 | 111 |
| 121 | 131 | 141 | 151 | 161 | 171 | 181 | 191 | 202 | 212 |
| 222 | 232 | 242 | 252 | 262 | 272 | 282 | 292 | 303 | 313 |
| 323 | 333 | 343 | 353 | 363 | 373 | 383 | 393 | 404 | 414 |
5. Armstrong Numbers
An Armstrong number is a special number where the sum of its digits, each raised to the power of the number of digits, equals the number itself! For example, 153 is special because 1³ + 5³ + 3³ = 153.
- Split the number into digits.
- Count how many digits it has.
- Raise each digit to that power and add them up.
- If the total equals the original number — it’s Armstrong!
- Only a few numbers are Armstrong — they're rare and cool!
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 153 |
| 370 | 371 | 407 | 1634 | 8208 | 9474 | 54748 | 92727 | 93084 | 548834 |
| 1741725 | 4210818 | 9800817 | 9926315 |
6. Harshad Numbers
Harshad numbers are numbers that can be divided by the sum of their digits. For example, 18 is Harshad because 1 + 8 = 9, and 18 ÷ 9 = 2.
- Add up the digits of the number.
- Try dividing the number by that sum.
- If there’s no remainder — it’s a Harshad number!
- Harshad means “joy-giver” in Sanskrit — easy to remember!
- Also called Niven numbers.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 12 | 18 | 20 | 21 | 24 | 27 | 30 | 36 | 40 | 42 |
| 45 | 48 | 50 | 54 | 60 | 63 | 70 | 72 | 80 | 81 |
| 84 | 90 | 100 | 102 | 108 | 111 | 112 | 114 | 117 | 120 |
| 126 | 132 | 133 | 135 | 140 | 144 | 150 | 152 | 153 | 156 |
7. Kaprekar Numbers
A Kaprekar number is a number where the square of the number can be split into two parts that add up to the original number. For example, 45² = 2025 → 20 + 25 = 45!
- Square the number.
- Split the result into two parts (left + right).
- Add the parts — if it equals the original, it's a Kaprekar number!
- They’re named after Indian mathematician D. R. Kaprekar.
- Left part can be 0!
| 1 | 9 | 45 | 55 | 99 | 297 | 703 | 999 | 2223 | 2728 |
| 4950 | 5050 | 7272 | 7777 | 9999 |
8. Amicable Numbers
Amicable numbers are a pair of numbers where each number is the sum of the proper divisors of the other. For example, 220 and 284 are amicable!
- Find all proper divisors of the first number.
- Add them up and see if they equal the second number.
- Do the same for the second number — it should give the first!
- They are like “best friends” in math!
- Often come in rare and larger pairs.
| 220 | 284 | 1184 | 1210 | 2620 | 2924 | 5020 | 5564 | 6232 | 6368 |
| 10744 | 10856 | 12285 | 14595 | 17296 | 18416 | 63020 | 76084 | 66928 | 66992 |
9. Deficient Numbers
Deficient numbers are numbers where the sum of their proper divisors is less than the number itself. For example, 10 has divisors 1, 2, and 5. Their sum is 8, which is less than 10.
- List the proper divisors (not including the number itself).
- Add them up.
- If the total is less than the number, it’s deficient!
- All prime numbers are deficient.
- Very common numbers!
| 1 | 2 | 3 | 4 | 5 | 7 | 8 | 9 | 10 | 11 |
| 13 | 14 | 15 | 16 | 17 | 19 | 21 | 22 | 23 | 25 |
| 26 | 27 | 28 | 29 | 31 | 32 | 33 | 34 | 35 | 36 |
10. Abundant Numbers
Abundant numbers are numbers where the sum of their proper divisors is more than the number. For example, 12 has divisors 1, 2, 3, 4, 6. Their sum is 16 — bigger than 12!
- Find all divisors except the number itself.
- Add them up — if it’s bigger than the number, it's abundant!
- First abundant number is 12.
- Most abundant numbers are even.
- Rare among small numbers!
| 12 | 18 | 20 | 24 | 30 | 36 | 40 | 42 | 48 | 54 |
| 56 | 60 | 66 | 70 | 72 | 78 | 80 | 84 | 88 | 90 |
🧠 Special Named Numbers
1. Triangular Numbers
Triangular numbers are numbers that can form an equilateral triangle. Each new number is the sum of the previous number and the next integer. For example, 6 is a triangular number because 1 + 2 + 3 = 6.
- Triangular numbers are the sum of the integers from 1 up to a certain number.
- Examples: 1, 3, 6, 10, 15.
- They can form a triangle when arranged in dots.
| 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 |
| 66 | 78 | 91 | 105 | 120 | 136 | 153 | 171 | 190 | 210 |
| 231 | 253 | 276 | 300 | 325 | 351 | 378 | 406 | 435 | 465 |
2. Square Numbers
Square numbers are numbers that can be made by multiplying a whole number by itself. For example, 9 is a square number because 3 × 3 = 9.
- Square numbers are perfect squares, like 1, 4, 9, 16, 25.
- They are formed by multiplying a number by itself.
- Examples: 1, 4, 9, 16, 25, 36, 49.
- Square numbers always have an odd number of divisors.
| 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 400 |
| 441 | 484 | 529 | 576 | 625 | 676 | 729 | 784 | 841 | 900 |
3. Cubic Numbers
Cubic numbers are numbers that can be made by multiplying a whole number by itself twice. For example, 8 is a cubic number because 2 × 2 × 2 = 8.
- Cubic numbers are the result of multiplying a number by itself three times.
- Examples: 1, 8, 27, 64, 125.
- They are formed by cubing the integers: n³.
- Perfect cubes are always positive numbers.
| 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
| 1331 | 1728 | 2197 | 2744 | 3375 | 4096 | 4913 | 5832 | 6859 | 8000 |
| 9261 | 10648 | 12167 | 13824 | 15747 | 17851 | 20076 | 22491 | 25000 | 27000 |
4. Fibonacci Numbers
Fibonacci numbers are a sequence where each number is the sum of the two preceding ones. It starts with 0 and 1, and then continues like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
- The sequence starts with 0 and 1.
- Each number is the sum of the two previous numbers.
- They appear in nature, such as in flower petals or the arrangement of leaves on a stem.
- The Fibonacci sequence grows quickly and has many interesting properties!
| 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 |
| 55 | 89 | 144 | 233 | 377 | 610 | 987 | 1597 | 2584 | 4181 |
| 6765 | 10946 | 17711 | 28657 | 46368 | 75025 | 121393 | 196418 | 317811 | 514229 |
5. Lucas Numbers
Lucas numbers are very similar to Fibonacci numbers, but they start with 2 and 1 instead of 0 and 1. They follow the same rule: each number is the sum of the two preceding ones.
- Lucas numbers are similar to Fibonacci numbers but start with 2 and 1.
- They appear in similar situations as Fibonacci numbers, like in nature and mathematics.
- They grow in the same way as Fibonacci numbers but with different starting points.
| 2 | 1 | 3 | 4 | 7 | 11 | 18 | 29 | 47 | 76 |
| 123 | 199 | 322 | 521 | 843 | 1364 | 2207 | 3571 | 5778 | 9349 |
| 15127 | 24476 | 39603 | 64079 | 103682 | 167761 | 271443 | 439204 | 710647 | 1149851 |
6. Tetrahedral Numbers
Tetrahedral numbers are like building a triangle in 3D! These numbers represent how many balls (or blocks) you would need to make a pyramid with a triangular base.
- The formula is: T(n) = n(n+1)(n+2)/6
- They are like 3D versions of triangle numbers.
- Start with 1 ball on top, then a triangle, then a bigger one, and so on!
- Use them to build stacks or pyramid shapes.
- Their shape is a tetrahedron — like a 3D triangle!
| 1 | 4 | 10 | 20 | 35 | 56 | 84 | 120 | 165 | 220 |
| 286 | 364 | 455 | 560 | 680 | 816 | 969 | 1140 | 1330 | 1540 |
7. Factorial Numbers
Factorial numbers come from multiplying a number by every smaller whole number. Written as n! — for example, 4! = 4 × 3 × 2 × 1 = 24!
- Factorial of 0 is 1 by rule!
- They grow really fast — huge numbers!
- Used in math puzzles, probability, and patterns.
- Each factorial includes all the ones before it!
- Handy for finding combinations and arrangements.
| 0! = 1 | 1! = 1 | 2! = 2 | 3! = 6 | 4! = 24 | 5! = 120 | 6! = 720 | 7! = 5040 | 8! = 40320 | 9! = 362880 |
| 10! = 3628800 | 11! = 39916800 | 12! = 479001600 | 13! = 6227020800 | 14! = 87178291200 | 15! = 1307674368000 |
🟦 Polygonal Numbers
1. Pentagonal Numbers
Pentagonal numbers are numbers that form a pattern like a pentagon. The first few pentagonal numbers are 1, 5, 12, 22, 35, and so on.
- Pentagonal numbers are generated by the formula (3n² − n) / 2.
- They grow in a pattern that can be shown as a pentagon made of dots.
- Examples: 1, 5, 12, 22, 35.
| 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 |
| 176 | 210 | 247 | 287 | 330 | 376 | 425 | 477 | 532 | 590 |
| 651 | 715 | 782 | 852 | 925 | 1001 | 1080 | 1162 | 1247 | 1335 |
2. Hexagonal Numbers
Hexagonal numbers are numbers that can be arranged in the shape of a hexagon. They follow a formula: n(2n − 1).
- Hexagonal numbers are the result of the formula n(2n − 1).
- They form a hexagon shape when visualized with dots.
- Examples: 1, 6, 15, 28, 45.
| 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 |
| 231 | 276 | 325 | 378 | 435 | 496 | 561 | 630 | 703 | 780 |
| 861 | 946 | 1035 | 1128 | 1225 | 1326 | 1431 | 1540 | 1653 | 1770 |
3. Heptagonal Numbers
Heptagonal numbers are numbers that can form a heptagon (a seven-sided shape). They follow the formula: n(5n − 3) / 2.
- Heptagonal numbers are formed using the formula n(5n − 3) / 2.
- They form a heptagon shape when visualized with dots.
- Examples: 1, 7, 18, 34, 55.
| 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 |
| 286 | 342 | 403 | 469 | 540 | 616 | 697 | 783 | 874 | 970 |
| 1071 | 1177 | 1288 | 1404 | 1525 | 1651 | 1782 | 1920 | 2064 | 2214 |
4. Octagonal Numbers
Octagonal numbers are numbers that can form an octagon (an eight-sided shape). They follow the formula: n(3n − 2).
- Octagonal numbers are created by the formula n(3n − 2).
- They form an octagon shape when visualized with dots.
- Examples: 1, 8, 21, 40, 65.
| 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 |
| 341 | 408 | 481 | 560 | 645 | 736 | 833 | 936 | 1045 | 1160 |
| 1281 | 1416 | 1557 | 1704 | 1857 | 2016 | 2181 | 2352 | 2539 | 2732 |
5. Nonagonal Numbers
Nonagonal numbers are numbers that can form a nonagon (a nine-sided shape). They follow the formula: n(7n − 5) / 2.
- Nonagonal numbers are created using the formula n(7n − 5) / 2.
- They form a nonagon shape when visualized with dots.
- Examples: 1, 9, 24, 46, 75.
| 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 |
| 396 | 474 | 560 | 654 | 756 | 866 | 984 | 1110 | 1244 | 1386 |
| 1536 | 1695 | 1862 | 2040 | 2226 | 2421 | 2625 | 2838 | 3060 | 3291 |
6. Decagonal Numbers
Decagonal numbers are numbers that can form a decagon (a ten-sided shape). They follow the formula: n(4n − 3).
- Decagonal numbers are created by the formula n(4n − 3).
- They form a decagon shape when visualized with dots.
- Examples: 1, 10, 35, 80, 150.
| 1 | 10 | 35 | 80 | 150 | 252 | 385 | 550 | 746 | 975 |
| 1235 | 1530 | 1860 | 2225 | 2625 | 3060 | 3530 | 4035 | 4575 | 5150 |
| 5760 | 6405 | 7085 | 7800 | 8550 | 9335 | 10155 | 10995 | 11880 | 12805 |
7. Hecagonal Numbers
Hecagonal numbers are numbers that can form a hecaton (a twelve-sided shape). They follow the formula: n(5n − 4).
- Hecagonal numbers are created by the formula n(5n − 4).
- They form a hecagon shape when visualized with dots.
- Examples: 1, 12, 35, 72, 133.
| 1 | 12 | 35 | 72 | 133 | 222 | 345 | 508 | 717 | 978 |
| 1287 | 1692 | 2197 | 2808 | 3531 | 4362 | 5307 | 6362 | 7525 | 8802 |
| 10103 | 11524 | 13071 | 14750 | 16567 | 18530 | 20645 | 22918 | 25355 | 27962 |
8. Tetragonal Numbers
Tetragonal numbers are numbers that can form a four-sided shape. They are created by the formula n(2n−1) for each number in the series.
- Tetragonal numbers form a pattern shaped like a square with dots.
- Each number is the sum of its predecessors.
- They grow in the same way as regular squares but follow their unique characteristics.
| 1 | 2 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 |
| 145 | 176 | 210 | 247 | 287 | 330 | 376 | 425 | 477 | 532 |
| 590 | 651 | 715 | 782 | 852 | 925 | 1001 | 1080 | 1162 | 1247 |
9. Centered Triangular Numbers
Centered triangular numbers make a triangle around a single center point! Imagine one dot in the middle, and more dots forming a triangle around it.
- Formula: C(n) = 3n(n−1)/2 + 1
- Starts with 1, then adds triangles around the center.
- Each step builds a bigger triangle.
- Common in dot patterns and art.
- Each level adds 3 more than the last layer.
| 1 | 4 | 10 | 19 | 31 | 46 | 64 | 85 | 109 | 136 |
| 166 | 199 | 235 | 274 | 316 | 361 | 409 | 460 | 514 | 571 |
10. Centered Square Numbers
Centered square numbers are made by placing a dot in the center and building layers in the shape of a square around it.
- Formula: C(n) = n² + (n−1)²
- Start with 1 in the center, then make bigger and bigger squares.
- Imagine rings around a center point forming a square.
- Used in tiling and pattern design.
- Always an odd number!
| 1 | 5 | 13 | 25 | 41 | 61 | 85 | 113 | 145 | 181 |
| 221 | 265 | 313 | 365 | 421 | 481 | 545 | 613 | 685 | 761 |
11. Centered Hexagonal Numbers
Centered hexagonal numbers make a hexagon shape (like a honeycomb!) around a single center point.
- Formula: C(n) = 3n(n−1) + 1
- Looks like a honeycomb with a center!
- Each ring adds 6 more than the one before it.
- Used in bee hives and tile designs.
- Think of six-sided shapes building out from a middle dot.
| 1 | 7 | 19 | 37 | 61 | 91 | 127 | 169 | 217 | 271 |
| 331 | 397 | 469 | 547 | 631 | 721 | 817 | 919 | 1027 | 1141 |
12. Centered Polygonal Numbers
Centered polygonal numbers form shapes like pentagons, hexagons, etc., around one center point. The shape depends on the number of sides!
- Formula: C(s,n) = (s−2)n(n−1)/2 + 1, where s = number of sides
- Each shape has its own unique pattern.
- Start with 1 in the middle, then build polygons around it.
- Works for triangles, squares, pentagons, and beyond!
- Explore centered shapes from triangles to decagons!
| 1 | 6 | 16 | 31 | 51 | 76 | 106 | 141 | 181 | 226 |
| 276 | 331 | 391 | 456 | 526 | 601 | 681 | 766 | 856 | 951 |
🟨 Combinatorial Sequences
1. Catalan Numbers
Catalan numbers count many types of combinatorial structures, such as valid ways to place parentheses, binary trees, and Dyck paths.
- Formula: C(n) = (2n)! / ((n+1)!n!)
- Used for counting valid combinations of parentheses.
- Important in combinatorics, like binary trees and paths.
- Starts with 1, then grows very quickly.
- Used in problems like how to organize pairs or arrange things.
| 1 | 2 | 5 | 14 | 42 | 132 | 429 | 1430 | 4862 | 16796 |
| 58786 | 208012 | 742900 | 2674440 | 9694845 | 35357670 | 129644790 | 477638700 | 1767263190 | 6564120420 |
2. Fibonacci Numbers
Fibonacci numbers are the sum of the two previous numbers, often seen in nature, such as in the spiral patterns of shells and flowers!
- Formula: F(n) = F(n-1) + F(n-2)
- Start with 0 and 1, then each new number is the sum of the previous two.
- Used to model things like growth, population, and paths.
- Common in nature — like the number of petals in flowers!
- Starts with 0, 1, and the numbers grow fast!
| 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 |
| 55 | 89 | 144 | 233 | 377 | 610 | 987 | 1597 | 2584 | 4181 |
3. Bell Numbers
Bell numbers count the number of ways to partition a set. They represent the number of ways to split a set into non-empty subsets.
- Bell numbers count how many different ways you can divide a set.
- Used in combinatorics to solve partitioning problems.
- They grow quickly as the set gets larger.
- They are important in counting combinations of groups.
- Starts with 1 for a set with one element.
| 1 | 2 | 5 | 15 | 52 | 203 | 877 | 4140 | 21147 | 115975 |
| 678570 | 4213597 | 27153233 | 185596241 | 1382958545 | 10480135630 | 82864864512 | 678570640395 | 5632640517664 | 47553835825777 |
4. Stirling Numbers
Stirling numbers count how many ways to partition a set into non-empty subsets. There are two types: first kind (ordered) and second kind (unordered).
- First kind counts ordered partitions, second kind counts unordered partitions.
- Used for counting how sets can be arranged into groups.
- Starts small but grows quickly!
- Important in problems involving groups and order.
| 1 | 1 | 6 | 15 | 52 | 203 | 877 | 4140 | 21147 | 115975 |
| 678570 | 4213597 | 27153233 | 185596241 | 1382958545 | 10480135630 | 82864864512 | 678570640395 | 5632640517664 | 47553835825777 |
5. Motzkin Numbers
Motzkin numbers count the number of paths on a grid that never go below the x-axis, with specific steps allowed.
- Formula: M(n) = (2n+1)(3n+1)/2
- They count paths that stay above a horizontal axis.
- Used in problems involving movement or geometry.
- Appears in combinatorics and graph theory.
- Can be seen as counting paths in specific shapes.
| 1 | 2 | 4 | 9 | 21 | 51 | 127 | 323 | 835 | 2185 |
| 5737 | 14905 | 38581 | 100029 | 258779 | 670497 | 1734007 | 4479941 | 11641395 | 30145649 |
6. Delannoy Numbers
Delannoy numbers count the number of paths from (0,0) to (m,n) in a grid using (1,0), (0,1), and (1,1) steps.
- Formula: D(m,n) = D(m-1, n) + D(m, n-1) + D(m-1, n-1)
- Used for counting paths in grid-based problems.
- Important in computer science and algorithms.
- Can be used to calculate number of ways to move across a grid.
| 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 |
| 66 | 78 | 91 | 105 | 120 | 136 | 153 | 171 | 190 | 210 |
7. Schröder Numbers
Schröder numbers count ways to insert parentheses in expressions, like Catalan numbers but with looser constraints.
- Formula: S(n) = 3S(n-1) + 4S(n-2) + 1
- Similar to Catalan numbers but less strict.
- Used in parsing and combinatorial geometry.
- Appears in the analysis of recursive functions.
| 1 | 2 | 6 | 22 | 90 | 402 | 1806 | 8062 | 36540 | 164384 |
| 742900 | 3352212 | 15041574 | 67613838 | 304472234 | 1371029862 | 6105310566 | 27420944166 | 122741407810 | 555213142850 |
8. Eulerian Numbers
Eulerian numbers count the number of permutations with a given number of ascents, often used in combinatorics and the study of permutations.
- Formula: A(n,k) = (−1)^k * (n choose k) * (n-k)! for counting ascents.
- Important for studying properties of permutations.
- Starts small but grows rapidly for larger values.
- Helps understand how elements can be rearranged in ordered sets.
| 1 | 1 | 3 | 11 | 35 | 126 | 462 | 1716 | 6435 | 24310 |
| 92378 | 343597 | 1302506 | 5011420 | 19383396 | 75251270 | 292389295 | 1135079818 | 4412384080 | 17336684910 |
9. Lah Numbers
Lah numbers count the ways to partition a set into ordered subsets. They are used in counting problems where order matters.
- Formula: L(n,k) = (n-1) choose (k-1) * (n-1)!
- Helps calculate the number of ordered partitions.
- Appears in combinatorics for ordered arrangements.
- Useful in problems involving ordered sequences.
| 1 | 3 | 15 | 105 | 945 | 10395 | 135135 | 2027025 | 34459425 | 654729075 |
10. Tribonacci Numbers
Tribonacci numbers are similar to Fibonacci numbers, but each term is the sum of the previous three terms.
- Formula: T(n) = T(n-1) + T(n-2) + T(n-3)
- Used in problems where each step depends on the previous three.
- Starts with 0, 1, and 1.
- Appears in sequences related to various physical models.
| 0 | 1 | 1 | 2 | 4 | 7 | 13 | 24 | 44 | 81 |
| 149 | 274 | 504 | 927 | 1701 | 3122 | 5747 | 10570 | 19439 | 35856 |
11. Tetrahedral Numbers
Tetrahedral numbers count 3D triangular arrangements, similar to triangular numbers in 2D.
- Formula: T(n) = n(n+1)(n+2)/6
- Appears in geometry for counting 3D shapes.
- Each new number represents a pyramid built of smaller tetrahedrons.
| 1 | 4 | 10 | 20 | 35 | 56 | 84 | 120 | 165 | 220 |
12. Partition Numbers
Partition numbers count the number of ways to write a given integer as a sum of positive integers, where the order of terms does not matter.
- Partitioning means breaking a number down into smaller parts, regardless of order.
- The number of partitions grows quickly as the number increases.
- Starts small, but the count can get large quickly!
- Common in number theory and combinatorics.
| 1 | 2 | 3 | 5 | 7 | 11 | 15 | 22 | 30 | 42 |
| 56 | 77 | 101 | 133 | 176 | 231 | 301 | 387 | 511 | 715 |
13. Narayana Numbers
Narayana numbers count specific structures, such as non-crossing partitions, which are partitions that avoid overlapping structures.
- Formula: N(n,k) = (n choose k) * (n-1 choose k-1)
- These numbers are used in combinatorics and partition theory.
- They count non-crossing partitions, useful in graph theory and lattice path problems.
- They appear in a variety of mathematical contexts, especially in counting problems.
| 1 | 2 | 3 | 5 | 7 | 11 | 15 | 22 | 30 | 42 |
| 56 | 77 | 101 | 133 | 176 | 231 | 301 | 387 | 511 | 715 |
14. Super Catalan Numbers
Super Catalan numbers are variants of Catalan numbers used for solving restricted counting problems, often appearing in combinatorics and combinatorial game theory.
- Formula: SC(n) = (2n choose n) - (2n choose n-1)
- Used in problems with specific constraints or limitations, such as balanced parentheses with more restrictions.
- Starts similarly to Catalan numbers, but diverges in later terms.
- Important in computational combinatorics and counting problems.
| 1 | 1 | 2 | 5 | 14 | 42 | 132 | 429 | 1430 | 4862 |
| 16796 | 58786 | 208012 | 742900 | 2674440 | 9694845 | 35357670 | 129644790 | 477638700 | 1767263190 |
15. Central Binomial Coefficients
Central binomial coefficients are the values in the center of Pascal’s Triangle. They are used in counting problems, such as paths in grids and combinatorics.
- Formula: C(2n, n) = (2n)! / (n!)²
- Used in problems involving binomial expansions and grid paths.
- Appears as the central numbers in Pascal’s Triangle.
- Important in combinatorics, algebra, and geometry.
| 1 | 2 | 6 | 20 | 70 | 252 | 924 | 3432 | 12870 | 48620 |
| 184756 | 705432 | 2704156 | 10400600 | 40116600 | 157773800 | 618702600 | 2443839376 | 9651637550 | 38062695750 |
16. Pascal’s Triangle
Pascal’s Triangle is a triangular array of binomial coefficients. Each number is the sum of the two numbers directly above it, and it has many applications in combinatorics and algebra.
- Each row represents the coefficients of binomial expansions (like (a+b)^n).
- The first and last numbers in each row are 1.
- The numbers are symmetric: the first half mirrors the second half.
- Used for problems involving combinations and algebraic expansions.
| 1 | 1 | 1 | 1 | 1 | |
| 1 | 2 | 1 | 3 | 3 | 1 |
| 1 | 3 | 3 | 1 |