A258133 Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.

1, 0, 0, 0, 2, -2, 2, 0, 0, 0, 3, -3, 3, 0, 0, 0, 4, -4, 4, 0, 0, 0, 5, -5, 5, 0, 0, 0, 6, -6, 6, 0, 0, 0, 7, -7, 7, 0, 0, 0, 8, -8, 8, 0, 0, 0, 9, -9, 9, 0, 0, 0, 10, -10, 10, 0, 0, 0, 11, -11, 11, 0, 0, 0, 12, -12, 12, 0, 0, 0, 13, -13, 13, 0, 0, 0, 14

Offset

0,5

Comments

This sequence is observed as the second difference in the expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^6)) (A029000), a sequence noted for its interlaced and structural coordination numbers.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,0,1,1,0,-1,-1).

Formula

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-6) + a(n-7) - a(n-9) - a(n-10). a(6*k-2) = -a(6*k-1) = a(6*k) = k+1 for k >= 1.

G.f.: (x^9-x^7-x^6+x^4-x^3+x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - Colin Barker, May 24 2015

a(n) = -a(-14-n) for all n in Z. - Michael Somos, Jun 07 2015

Example

G.f. = 1 + 2*x^4 - 2*x^5 + 2*x^6 + 3*x^10 - 3*x^11 + 3*x^12 + 4*x^16 + ...

Mathematica

a[ n_] := With[ {m=n-1}, If[ OddQ[ Quotient[ m, 3]], Quotient[ m+9, 6] (-1)^Mod[m, 3], 0]]; (* Michael Somos, Jun 07 2015 *)

Prog

(PARI) Vec((x^9-x^7-x^6+x^4-x^3+x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, May 24 2015
(PARI) {a(n) = n--; if(n\3%2, (n+9)\6 * (-1)^(n%3), 0)}; /* Michael Somos, Jun 07 2015 */

Crossrefs

Sequence in context: A037865 A039969 A039967 * A327152 A123186 A127323

Adjacent sequences: A258130 A258131 A258132 * A258134 A258135 A258136

Keyword

sign,easy

Author

Avi Friedlich, May 21 2015

Status

approved