A057587 Nonnegative numbers of form n*(n^2+-1)/2.

0, 1, 3, 5, 12, 15, 30, 34, 60, 65, 105, 111, 168, 175, 252, 260, 360, 369, 495, 505, 660, 671, 858, 870, 1092, 1105, 1365, 1379, 1680, 1695, 2040, 2056, 2448, 2465, 2907, 2925, 3420, 3439, 3990, 4010, 4620, 4641, 5313, 5335, 6072, 6095, 6900, 6924, 7800

Offset: 0

Author

N. J. A. Sloane, Oct 05 2000

Keywords

• nonn
• easy

Comments

Taking alternate terms gives A027480 and A006003. - Jeremy Gardiner, Apr 10 2005

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A027480, A006003.

Programs

Magma

[(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Mar 27 2015

Maple

A057587:=n->(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32: seq(A057587(n), n=0..50); # Wesley Ivan Hurt, Mar 27 2015

Mathematica

CoefficientList[Series[x*(1 + 2 x - x^2 + x^3)/((x - 1)^4*(x + 1)^3), {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 27 2015 *)
Table[(2 n^3 + 9 n^2 + 15 n + 5 + (3 n^2 + n - 5) (-1)^n) / 32, {n, 0, 50}] (* Vincenzo Librandi, Mar 28 2015 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,3,5,12,15,30},50] (* Harvey P. Dale, Nov 29 2022 *)

PARI

concat(0, Vec(x*(x^3-x^2+2*x+1)/((x-1)^4*(x+1)^3) + O(x^100))) \\ Colin Barker, Nov 18 2014

Formula

a(n) = (2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32. - Luce ETIENNE, Nov 18 2014

From Wesley Ivan Hurt, Mar 27 2015: (Start)

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

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). (End)

a(2*n+1) + a(2*n) = (a(2*n+1) - a(2*n))^3. - Greg Dresden, Feb 10 2022

a(2*n+1)^2 - a(2*n)^2 = (n+1)^4. - Adam Michael Bere, Feb 15 2024