A121470 Expansion of x*(1+5*x+2*x^2+x^3)/((1+x)*(1-x)^3).

1, 7, 16, 31, 49, 73, 100, 133, 169, 211, 256, 307, 361, 421, 484, 553, 625, 703, 784, 871, 961, 1057, 1156, 1261, 1369, 1483, 1600, 1723, 1849, 1981, 2116, 2257, 2401, 2551, 2704, 2863, 3025, 3193, 3364, 3541, 3721, 3907, 4096, 4291, 4489, 4693, 4900

Offset

1,2

Links

Table of n, a(n) for n=1..47.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

From R. J. Mathar, Jul 10 2009: (Start)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) = 5/8 - 3n/2 + 9n^2/4 + 3*(-1)^n/8.

G.f.: x*(1+5*x+2*x^2+x^3)/((1+x)*(1-x)^3). (End)

a(n) = (18*n^2 - 12*n + 5 + 3*(-1)^n)/8. - Charles R Greathouse IV, Jun 03 2026

Maple

A121410 := proc(nmin) local M, a, v, wev, wod, n ; a := [] ; M := linalg[matrix](2, 2, [0, 1, -1, 2]) ; v := linalg[vector](2, [1, 7]) ; wev := linalg[vector](2, [0, 3]) ; wod := linalg[vector](2, [0, 6]) ; while nops(a) < nmin do a := [op(a), v[1]] ; n := nops(a)+1 ; v := evalm(M &* v) ; if n mod 2 = 0 then v := evalm(v+wev) ; else v := evalm(v+wod) ; fi ; od: RETURN(a) ; end: A121410(80) ; # R. J. Mathar, Sep 18 2007

Mathematica

M := {{0, 1}, {-1, 2} } v[1] = {1, 7} w[n_] = If[Mod[n, 2] == 0, {0, 3}, {0, 6}] v[n_] := v[n] = M.v[n - 1] + w[n] a = Table[v[n][[1]], {n, 1, 30}]
CoefficientList[Series[x (1+5x+2x^2+x^3)/((1+x)(1-x)^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 7, 16, 31}, 50] (* Harvey P. Dale, Mar 10 2017 *)

Prog

(PARI) a(n)=(18*n^2-12*n+5)\/8 \\ Charles R Greathouse IV, Jun 03 2026

Crossrefs

Cf. A003215, A005448.

Sequence in context: A140511 A286119 A286085 * A286733 A286773 A019541

Adjacent sequences: A121467 A121468 A121469 * A121471 A121472 A121473

Keyword

nonn,easy

Author

Roger L. Bagula, Sep 07 2006

Extensions

Edited by N. J. A. Sloane, Sep 16 2006

More terms from R. J. Mathar, Sep 18 2007

New name from Joerg Arndt, Jun 28 2013

Status

approved