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A022106 Fibonacci sequence beginning 1, 16. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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).

a(n) = 16*A000045(n) + A000045(n-1). [Paolo P. Lava, May 19 2015]

MAPLE

with(numtheory): with(combinat): P:=proc(q) local n;

for n from 0 to q do print(16*fibonacci(n)+fibonacci(n-1));

od; end: P(30); # Paolo P. Lava, May 19 2015

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A244431 A151977 A252492 * A041518 A042195 A041520

Adjacent sequences:  A022103 A022104 A022105 * A022107 A022108 A022109

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 23 20:15 EDT 2017. Contains 283957 sequences.