A097038 A Jacobsthal variant.

0, 1, 1, 5, 7, 21, 35, 85, 155, 341, 651, 1365, 2667, 5461, 10795, 21845, 43435, 87381, 174251, 349525, 698027, 1398101, 2794155, 5592405, 11180715, 22369621, 44731051, 89478485, 178940587, 357913941, 715795115, 1431655765, 2863245995

Offset: 0

Author

Paul Barry, Jul 19 2004

Keywords

• easy
• nonn

Comments

Convolution of A001045 and A077957.

Also interleaving of A002450(n+1) and A006095(n+1).

Links

• Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Crossrefs

Cf. A001045, A077957.

Cf. A002450, A006095.

Programs

PARI

concat(0, Vec(x/((1-2*x^2)*(1-x-2*x^2)) + O(x^50))) \\ Michel Marcus, Nov 13 2015

PARI

vector(50, n, n--; 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2) \\ Altug Alkan, Nov 13 2015

Formula

G.f.: 1/(1-x-2*x^2) - 1/(1-2*x^2) = x/((1-2*x^2)*(1-x-2*x^2));

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

a(n) = sum{k=0..floor((n+1)/2), binomial(n-k+1, k-1)2^k };

a(n) = sum{k=0..n, 2^(k/2)(1+(-1)^k)A001045(n-k)/2 };

a(n) = A001045(n+1)-A077957(n).