A158613 Expansion of (1 - 2*x^3 - x^4 - x^5 + x^6 + x^7 - x^8)/(1 - x^3)^3.

1, 0, 0, 1, -1, -1, 1, -2, -4, 1, -3, -9, 1, -4, -16, 1, -5, -25, 1, -6, -36, 1, -7, -49, 1, -8, -64, 1, -9, -81, 1, -10, -100, 1, -11, -121, 1, -12, -144, 1, -13, -169, 1, -14, -196, 1, -15, -225, 1, -16, -256, 1, -17, -289, 1, -18, -324, 1, -19, -361, 1, -20, -400, 1, -21, -441

Offset

0,8

Comments

The sequence is given by the successive triples (1, -m, -m^2) with m = 0, 1, 2, 3, ... - Bruno Berselli, Aug 23 2018

Links

Table of n, a(n) for n=0..65.

Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Formula

From Bruno Berselli, Aug 23 2018: (Start)

G.f.: (1 - 2*x^3 - x^4 - x^5 + x^6 + x^7 - x^8)/((1 - x)^3*(1 + x + x^2)^3).

a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>8.

a(n) = -(-1)^((n+1) mod 3)*floor(n/3)^(n mod 3). (End)

Example

As array:
1,   0,    0;
1,  -1,   -1;
1,  -2,   -4;
1,  -3,   -9;
1,  -4,  -16;
1,  -5,  -25;
1,  -6,  -36;
1,  -7,  -49;
1,  -8,  -64;
1,  -9,  -81;
1, -10, -100 etc.

Mathematica

CoefficientList[Series[(1-2x^3-x^4-x^5+x^6+x^7-x^8)/(1-x^3)^3, {x, 0, 100}], x] (* or *) LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 0, 0, 1, -1, -1, 1, -2, -4}, 100] (* Harvey P. Dale, Nov 22 2021 *)

Prog

(Magma) &cat [[1, -n, -n^2]: n in [0..25]]; // Bruno Berselli, Aug 23 2018

Crossrefs

Cf. A000463, A156133, A234357.

Sequence in context: A208526 A275895 A369292 * A360859 A209573 A100075

Adjacent sequences: A158610 A158611 A158612 * A158614 A158615 A158616

Keyword

sign,tabf,easy,less

Author

Roger L. Bagula, Mar 22 2009

Extensions

Edited, new name, and a(1)-a(2) added by Bruno Berselli, Aug 23 2018

Status

approved