A024598 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = (odd natural numbers).

1, 3, 14, 22, 55, 73, 140, 172, 285, 335, 506, 578, 819, 917, 1240, 1368, 1785, 1947, 2470, 2670, 3311, 3553, 4324, 4612, 5525, 5863, 6930, 7322, 8555, 9005, 10416, 10928, 12529, 13107, 14910, 15558, 17575, 18297, 20540, 21340, 23821, 24703, 27434, 28402

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Sum of the areas of all rectangles with odd side lengths r and s such that r + s = 2n. - Wesley Ivan Hurt, Apr 21 2020

Links

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

Crossrefs

Cf. A143218.

Programs

Magma

[(4*n^3 +3*n^2 +2*n -3*n^2*(-1)^n)/12: n in [1..50]]; // G. C. Greubel, Jul 12 2022

Mathematica

LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,3,14,22,55,73,140},50] (* Harvey P. Dale, Aug 30 2021 *)

PARI

Vec(x*(3*x^4+2*x^3+8*x^2+2*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 17 2015

PARI

odd(n) = 2*n-1;
a(n) = sum(j=1, (n+1)\2, odd(j)*odd(n+1-j)); \\ Michel Marcus, Jan 17 2015

SageMath

[(4*n^3 +3*n^2 +2*n -3*n^2*(-1)^n)/12 for n in (1..50)] # G. C. Greubel, Jul 12 2022

Formula

a(n) = (4*n^3 + 3*n^2 + 2*n - 3*n^2*(-1)^n)/12. - Luce ETIENNE, Jan 17 2015

G.f.: x*(3*x^4 + 2*x^3 + 8*x^2 + 2*x + 1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 17 2015

a(n) = Sum_{i=1..n} i * (2*n-i) * (i mod 2). - Wesley Ivan Hurt, Apr 21 2020

E.g.f.: (x/6)*( 2*(3 + 3*x + x^2)*cosh(x) + (3 + 9*x + 2*x^2)*sinh(x) ). - G. C. Greubel, Jul 12 2022