A111913 Expansion of x*(-2-3*x-x^2+x^7+x^8+2*x^4) / ((x-1)*(x+1)*(x^8-x^4+1)).

0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3, 3, 3, 6, 4, 5, 1, 5, 1, 4, -2, 1, -3, 1, -3, -2, -4, -1, -1, -1, -1, 0, 2, 3, 3

Offset: 0

Author

Creighton Dement, Aug 20 2005

Keywords

• easy
• sign

Comments

It appears that (a(n)) has period 24.

The above conjecture is correct, since x^24 = 1 mod (x-1)*(x+1)*(x^8-x^4+1). - Charles R Greathouse IV, Feb 07 2013

Floretion Algebra Multiplication Program, FAMP Code: 4ibasesigcycsumseq[ + .5'i + .5j' + .5'ij' + .5e], sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code); apart from initial term 0.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1,0,-1,0,1).

Crossrefs

Cf. A111912, A111914, A111915, A085846.

Programs

Mathematica

LinearRecurrence[{0,1,0,1,0,-1,0,-1,0,1},{0,2,3,3,3,3,6,4,5,1},120] (* Harvey P. Dale, Apr 14 2019 *)

PARI

a(n)=[0,2,3,3,3,3,6,4,5,1,5,1,4,-2,1,-3,1,-3,-2,-4,-1,-1,-1,-1][n%24+1] \\ Charles R Greathouse IV, Feb 07 2013

PARI

concat(0, Vec(x*(2 + 3*x + x^2 - 2*x^4 - x^7 - x^8) / ((1 - x)*(1 + x)*(1 - x^4 + x^8)) + O(x^80))) \\ Colin Barker, May 18 2019

Formula

a(n) = a(n-2) + a(n-4) - a(n-6) - a(n-8) + a(n-10) for n>9. - Colin Barker, May 18 2019