A022106 Fibonacci sequence beginning 1, 16.

1, 16, 17, 33, 50, 83, 133, 216, 349, 565, 914, 1479, 2393, 3872, 6265, 10137, 16402, 26539, 42941, 69480, 112421, 181901, 294322, 476223, 770545, 1246768, 2017313, 3264081, 5281394, 8545475, 13826869

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n-1)=sum(P(16;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=15. These are the SW-NE diagonals in P(16;n,k), the (16,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

a(n) = A109754(15, n+1) = A101220(15, 0, n+1).

Programs

Magma

a0:=1; a1:=16; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

Mathematica

a={};b=1;c=16;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1,1},{1,16},40] (* Harvey P. Dale, Jun 22 2016 *)

Formula

a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=16. a(-1):=15.

G.f.: (1+15*x)/(1-x-x^2).