A094968 - OEIS

%I #14 Sep 29 2017 10:58:58

%S 1,4,7,14,25,48,91,178,349,692,1375,2742,5473,10936,21859,43706,87397,

%T 174780,349543,699070,1398121,2796224,5592427,11184834,22369645,

%U 44739268,89478511,178956998,357913969,715827912,1431655795,2863311562

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

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

%C 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]

%H Colin Barker, <a href="/A094968/b094968.txt">Table of n, a(n) for n = 0..1000</a>

%H D. H. Lehmer, <a href="https://www.jstor.org/stable/2299356">On Stern's Diatomic Series</a>, Amer. Math. Monthly 36(1) 1929, pp. 59-67.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).

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

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

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

%F From _Colin Barker_, Sep 29 2017: (Start)

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

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

%F (End)

%o (PARI) Vec((1 + x - 4*x^2) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^30)) \\ _Colin Barker_, Sep 29 2017

%Y Cf. A000045, A001045, A049456, A128209.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 26 2004