N-bonacci numbers

This article page is a stub, please help by expanding it.

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

${\begin{array}{rcl}a(n)&:=&a_{n},\quad 0\leq n\leq N-2;\\a(N-1)&:=&a_{N-1};\end{array}}$

instead of two initial terms, and where each subsequent term is the sum of the previous $\scriptstyle N$ terms

$a(n):=\sum _{i=n-N}^{n-1}a(i),\quad n\geq N.$

The most common choices for the $\scriptstyle N$ initial terms are either all 0 or all 1 for the first $\scriptstyle N-1$ initial terms $a_{n}$, $\scriptstyle 0\,\leq \,n\,\leq \,N-2$, and 1 for the $\scriptstyle N$th initial term $a_{N-1}$.

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 $\scriptstyle a(n)\,=\,0,\,0\,\leq \,n\,\leq \,N-2;\;a(N-1)\,=\,1$. Each subsequent term is the sum of the previous $\scriptstyle N$ 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 $\scriptstyle N$-bonacci numbers $\scriptstyle (a(n)\,=\,0,\,0\,\leq \,n\,\leq \,N-2;\;a(N-1)\,=\,1)$

$N$	$\scriptstyle N$-bonacci numbers sequences $\scriptstyle F_{n}^{(N)},\,n\,\geq \,0.$	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

• 0-bonacci numbers (medenacci numbers) (degenerate: no initial term, empty sum yields all 0's sequence)
• 1-bonacci numbers (enanacci numbers) (degenerate: one initial term, constant sequence)
• 2-bonacci numbers (Fibonacci numbers)
• 3-bonacci numbers (tribonacci numbers)
• 4-bonacci numbers (tetranacci numbers)
• 5-bonacci numbers (pentanacci numbers)
• 6-bonacci numbers (hexanacci numbers)
• 7-bonacci numbers (heptanacci numbers)
• 8-bonacci numbers (octanacci numbers)
• 9-bonacci numbers (enneanacci numbers)
• 10-bonacci numbers (decanacci numbers)
• 11-bonacci numbers (hendecanacci numbers)
• 12-bonacci numbers (dodecanacci numbers)