cp's OEIS Frontend

A097038 A Jacobsthal variant.

%I A097038 #12 Nov 13 2015 21:44:38
%S A097038 0,1,1,5,7,21,35,85,155,341,651,1365,2667,5461,10795,21845,43435,
%T A097038 87381,174251,349525,698027,1398101,2794155,5592405,11180715,22369621,
%U A097038 44731051,89478485,178940587,357913941,715795115,1431655765,2863245995
%N A097038 A Jacobsthal variant.
%C A097038 Convolution of A001045 and A077957.
%C A097038 Also interleaving of A002450(n+1) and A006095(n+1).
%H A097038 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-4).
%F A097038 G.f.: 1/(1-x-2*x^2) - 1/(1-2*x^2) = x/((1-2*x^2)*(1-x-2*x^2));
%F A097038 a(n) = 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2;
%F A097038 a(n) = sum{k=0..floor((n+1)/2), binomial(n-k+1, k-1)2^k };
%F A097038 a(n) = sum{k=0..n, 2^(k/2)(1+(-1)^k)A001045(n-k)/2 };
%F A097038 a(n) = A001045(n+1)-A077957(n).
%o A097038 (PARI) concat(0, Vec(x/((1-2*x^2)*(1-x-2*x^2)) + O(x^50))) \\ _Michel Marcus_, Nov 13 2015
%o A097038 (PARI) vector(50, n, n--; 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2) \\ _Altug Alkan_, Nov 13 2015
%Y A097038 Cf. A001045, A077957.
%Y A097038 Cf. A002450, A006095.
%K A097038 easy,nonn
%O A097038 0,4
%A A097038 _Paul Barry_, Jul 19 2004