A140153 a(1)=1, a(n) = a(n-1) + n^3 if n odd, a(n) = a(n-1) + n^1 if n is even.

1, 3, 30, 34, 159, 165, 508, 516, 1245, 1255, 2586, 2598, 4795, 4809, 8184, 8200, 13113, 13131, 19990, 20010, 29271, 29293, 41460, 41484, 57109, 57135, 76818, 76846, 101235, 101265, 131056, 131088, 167025, 167059, 209934, 209970, 260623

Offset: 1

Author

Artur Jasinski, May 12 2008

Keywords

• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

GAP

a:=[1];; for n in [2..40] do a[n]:=a[n-1]+((1-(-1)^n)/2)*n^3+((1+(-1)^n)/2)*n; od; a; # Muniru A Asiru, Jul 12 2018

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5))); // G. C. Greubel, Jul 12 2018

Maple

a:=proc(n) option remember: if n=1 then 1 elif modp(n,2)<>0 then procname(n-1)+n^3 else procname(n-1)+n; fi: end; seq(a(n),n=1..40); # Muniru A Asiru, Jul 12 2018

Mathematica

a = {}; r = 3; s = 1; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
CoefficientList[Series[x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5), {x,0,30}], x] (* G. C. Greubel, Jul 12 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^3,a+n+1]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,3,30,34,159,165,508,516,1245},40] (* Harvey P. Dale, Aug 26 2021 *)

PARI

x='x+O('x^30); Vec(x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5)) \\ G. C. Greubel, Jul 12 2018

Formula

a(n) = a(n-1) + {[1-(-1)^n]/2}*n^3 + {[1+(-1)^n]/2}*n, with a(1)=1.

From R. J. Mathar, Feb 22 2009: (Start)

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).

G.f.: x*(1+2*x+23*x^2-4*x^3+23*x^4+2*x^5+x^6)/((1+x)^4*(1-x)^5). (End)