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

1,2

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).

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

Mathematica

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

Prog

(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
(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
(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

Crossrefs

Cf. A143218.

Sequence in context: A255219 A226341 A024473 * A019001 A034103 A034113

Adjacent sequences: A024595 A024596 A024597 * A024599 A024600 A024601

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved