A087463 Generalized multiplicative Jacobsthal sequence.

0, 1, 1, 0, 5, 11, 0, 43, 85, 0, 341, 683, 0, 2731, 5461, 0, 21845, 43691, 0, 174763, 349525, 0, 1398101, 2796203, 0, 11184811, 22369621, 0, 89478485, 178956971, 0, 715827883, 1431655765, 0, 5726623061, 11453246123, 0, 45812984491, 91625968981, 0

Offset: 0

Author

Paul Barry, Sep 08 2003

Keywords

• easy
• nonn

Comments

Set A001045(3n)=0 in A001045.

2^n = A087462(n) + a(n) + A087464(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).

Programs

PARI

concat(0, Vec(-x*(4*x^4-2*x^3+x+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)=1, 1, 0)*C(n, k).

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

From Colin Barker, Nov 02 2015: (Start)

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

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

(End)