A087464 Generalized multiplicative Jacobsthal sequence.

0, 0, 2, 0, 6, 10, 0, 42, 86, 0, 342, 682, 0, 2730, 5462, 0, 21846, 43690, 0, 174762, 349526, 0, 1398102, 2796202, 0, 11184810, 22369622, 0, 89478486, 178956970, 0, 715827882, 1431655766, 0, 5726623062, 11453246122, 0, 45812984490, 91625968982, 0

Offset: 0

Author

Paul Barry, Sep 08 2003

Keywords

• easy
• nonn

Comments

2^n = A087462(n) + A087463(n) + a(n) provides a decomposition of Pascal's triangle.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,8).

Crossrefs

Cf. A001045, A087462, A087463.

Programs

Mathematica

LinearRecurrence[{0,0,7,0,0,8},{0,0,2,0,6,10},40] (* Harvey P. Dale, Aug 31 2015 *)

PARI

concat(vector(2), Vec(2*x^2*(2*x^3-3*x^2-1)/((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)) + O(x^100))) \\ Colin Barker, Nov 02 2015

Formula

a(n) = Sum_{k=0..n} if(mod(n*k, 3)=2, 1, 0) * C(n, k).

a(n) = (2/9)*(2^n-3*0^n+2*(-1)^n)*(1-cos(2*Pi*n/3)).

From Colin Barker, Nov 02 2015: (Start)

a(n) = 7*a(n-3)+8*a(n-6) for n>5.

G.f.: 2*x^2*(2*x^3-3*x^2-1) / ((x+1)*(2*x-1)*(x^2-x+1)*(4*x^2+2*x+1)).

(End)