A220493 Fibonacci 15-step numbers, a(n) = a(n-1) + a(n-2) + ... + a(n-15).

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32767, 65533, 131064, 262124, 524240, 1048464, 2096896, 4193728, 8387328, 16774400, 33548288, 67095552, 134189056, 268374016, 536739840, 1073463296, 2146893825, 4293722117, 8587313170

Offset: 1

Author

Ruskin Harding, Feb 20 2013

Keywords

• nonn
• easy

Comments

Also called Pentadecanacci numbers. In previous similar sequences, a(1), ..., a(n-1) have been set equal to zero and a(n)=1. For example, A168084 (Fibonacci 13-step numbers) has 12 0's as the first 12 terms and a(13)=1.

Links

• Robert Israel, Table of n, a(n) for n = 1..3320
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
• Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Crossrefs

Cf. A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), A001591 (pentanacci).

Programs

Maple

f:= gfun:-rectoproc({a(n) = add(a(n-i),i=1..15), seq(a(n)=0,n=-14..0),a(1)=1},a(n),remember):
map(f, [$1..100]); # Robert Israel, Feb 19 2019

Mathematica

FibonacciSequence[n_, kMax_] := Module[{a, s}, a = Join[{1}, Table[0, {n - 1}]]; lst = {}; Table[s = Plus @@ a; a = RotateLeft[a]; a[[n]] = s, {k, 1, kMax}]]; FibonacciSequence[15, 50] (* T. D. Noe, Feb 20 2013 *)

Formula

G.f.: x/(1-Sum_{k=1..15} x^k). - Robert Israel, Feb 19 2019