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

Author

Roger L. Bagula, Sep 07 2006

Keywords

• nonn

Links

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

Crossrefs

Cf. A003215, A005448.

Programs

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 *)

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)

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