Fibonacci n-step number sequences
These number series are an expansion of the ordinary Fibonacci sequence where:
- For $n = 2$ we have the Fibonacci sequence; with initial values $[1, 1]$ and $F_k^2 = F_{k-1}^2 + F_{k-2}^2$
- For $n = 3$ we have the tribonacci sequence; with initial values $[1, 1, 2]$ and $F_k^3 = F_{k-1}^3 + F_{k-2}^3 + F_{k-3}^3$
- For $n = 4$ we have the tetranacci sequence; with initial values $[1, 1, 2, 4]$ and $F_k^4 = F_{k-1}^4 + F_{k-2}^4 + F_{k-3}^4 + F_{k-4}^4$...
- For general $n>2$ we have the Fibonacci $n$-step sequence - $F_k^n$; with initial values of the first $n$ values of the $(n-1){{description}}#39;th Fibonacci $n$-step sequence $F_k^{n-1}$; and $k{{description}}#39;th value of this $n{{description}}#39;th sequence being $F_k^n = \sum_{i=1}^{(n)} {F_{k-i}^{(n)}}$
For small values of $n$, Greek numeric prefixes are sometimes used to individually name each series.
Fibonacci $n$-step sequences:
$n$ | Series name | Values |
---|---|---|
2 | fibonacci | 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... |
3 | tribonacci | 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... |
4 | tetranacci | 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... |
5 | pentanacci | 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... |
6 | hexanacci | 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... |
7 | heptanacci | 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... |
8 | octonacci | 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... |
9 | nonanacci | 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... |
10 | decanacci | 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ... |
Allied sequences can be generated where the initial values are changed: The Lucas series sums the two preceding values like the fibonacci series for $n=2$ but uses $[2, 1]$ as its initial values.
Test
{{test}}Console output