A136047 a(1)=1, a(n)=a(n-1)+n if n even, a(n)=a(n-1)+n^2 if n is odd.

1, 3, 12, 16, 41, 47, 96, 104, 185, 195, 316, 328, 497, 511, 736, 752, 1041, 1059, 1420, 1440, 1881, 1903, 2432, 2456, 3081, 3107, 3836, 3864, 4705, 4735, 5696, 5728, 6817, 6851, 8076, 8112, 9481, 9519, 11040, 11080, 12761, 12803, 14652, 14696, 16721

Offset: 1

Author

Zak Seidov, Dec 12 2007

Keywords

• nonn
• easy

Comments

The only prime terms are 3, 41, 47.

The semiprime terms are A136048.

Cf. A001082/A135370: f(1) = 1, then if n even/odd f(n) = n+f(n-1), if n odd/even f(n) = 2*n+f(n-1).

Links

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

Crossrefs

Cf. A001082, A135370, A136048.

Programs

Mathematica

a[1]=1;a[n_]:=a[n]=a[n-1]+n^(1+Mod[n,2]); Table[a[n],{n,100}]
nxt[{n_,a_}]:={n+1,If[OddQ[n],a+n+1,a+(n+1)^2]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* Harvey P. Dale, Oct 11 2015 *)

Formula

a(n) = (1/12)(1 + n)(2n^2+7n-3) if n is odd, a(n)=(1/12)n(2n^2+3n+4) if n is even.

a(n) = (-3 + 3*(-1)^n + 8*n + 12*n^2 - 6*(-1)^n*n^2 + 4*n^3)/24.

a(1)=1 then a(n) = a(n-1)+n^(if n is even then 1 else 2),

or a(n) = a(n-1)+n^(1+mod(n,2)),

or a(n) = a(n-1)+n^((3-(-1)^n)/2).

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

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

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

Extensions

Edited by Michel Marcus, Mar 02 2022