A094968 Indices of Fibonacci numbers in Stern's diatomic series A049456 regarded as a single linear sequence.

1, 4, 7, 14, 25, 48, 91, 178, 349, 692, 1375, 2742, 5473, 10936, 21859, 43706, 87397, 174780, 349543, 699070, 1398121, 2796224, 5592427, 11184834, 22369645, 44739268, 89478511, 178956998, 357913969, 715827912, 1431655795, 2863311562

Offset: 0

Author

Paul Barry, May 26 2004

Keywords

• easy
• nonn

Comments

By definition, A049456(a(n))=Fib(n+2).

The rank of Fib(n+2) in row n of A049456 (regarded as an irregular triangle read by rows) is A128209(n) = A001045(n)+1. [Comment edited by N. J. A. Sloane, Nov 23 2016]

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Crossrefs

Cf. A000045, A001045, A049456, A128209.

Programs

PARI

Vec((1 + x - 4*x^2) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^30)) \\ Colin Barker, Sep 29 2017

Formula

G.f. : (1+x-4*x^2) / ((1-x)*(1-x^2)*(1-2*x)).

a(n) = 2^n + n + Jacobsthal(n).

a(n) = A006127(n) + A001045(n).

From Colin Barker, Sep 29 2017: (Start)

a(n) = ((-1)^(1+n) + 2^(2+n) + 3*n) / 3.

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4) for n>3.

(End)