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, 5726623061, 11453115051

Offset

0,4

Comments

Convolution of A001045 and A077957.

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

Links

Table of n, a(n) for n=0..34.

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

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).

E.g.f.: exp(-x)/3 + 2*exp(2*x)/3 - exp(-sqrt(2)*x)/2 - exp(sqrt(2)*x)/2. - Amiram Eldar, Jan 15 2026

Mathematica

LinearRecurrence[{1, 4, -2, -4}, {0, 1, 1, 5}, 50] (* Amiram Eldar, Jan 15 2026 *)

Prog

(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

Crossrefs

Cf. A001045, A002450, A006095, A077957.

Sequence in context: A076197 A002596 A098597 * A049114 A179189 A030735

Adjacent sequences: A097035 A097036 A097037 * A097039 A097040 A097041

Keyword

easy,nonn

Author

Paul Barry, Jul 19 2004

Status

approved