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N-bonacci numbers

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The N-bonacci numbers arise from a recurrence relation like that of the Fibonacci numbers, but with initial terms defined as

instead of two initial terms, and where each subsequent term is the sum of the previous terms

The most common choices for the initial terms are either all 0 or all 1 for the first initial terms , , and 1 for the th initial term .

N-bonacci numbers with first N-1 initial terms set to 0 and Nth initial term set to 1

The N-bonacci numbers with initial terms set to . Each subsequent term is the sum of the previous terms. For example, 47-bonacci numbers use a recurrence relation with 46 initial 0's and one 1, and each subsequent term is the sum of the previous 47 terms.

Table of -bonacci numbers
-bonacci numbers sequences A-number
0 {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...} A000004
1 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012
2 {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, 832040, ...} A000045
3 {0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, ...} A000073
4 {0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, ...} A000078
5 {0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, ...} A001591
6 {0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, ...} A001592
7 {0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, ...} A122189
8 {0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, ...} A079262
9 {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, ...} A104144
10 {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521472, 1042432, 2083841, 4165637, 8327186, ...} A122265
11 {}  
12 {}  

N-bonacci numbers with all N initial terms set to 1

  1.  
  2.  
  3. A000213 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.
  4. A000288 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0)=a(1)=a(2)=a(3)=1.
  5. A000322 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) with a(0)=a(1)=a(2)=a(3)=a(4)=1.
  6.  
  7.  
  8.  
  9. A127193 A 9th order Fibonacci sequence with a(1)=...=a(9)=1.
  10. A127194 A 10th order Fibonacci sequence with a(1)=...=a(10)=1.
  11. A127624 An 11th order Fibonacci sequence. a(n) = a(n-1) + ... + a(n-11) with a(1)=...=a(11)=1.
  12. A207539 Dodecanacci numbers (12th-order Fibonacci sequence): a(n) = a(n-1) +...+ a(n-12) with a(0)=...=a(11)=1.
  13. A163551 13th order Fibonacci numbers: a(n) = a(n-1) +...+ a(n-13) with a(1)=...=a(13)=1.

N-bonacci numbers with N initial terms set to other values

  • A001630 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=a(1)=0, a(2)=1, a(3)=2.

See also

Notes