A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360.

0, 1, 9, 43, 147, 406, 966, 2058, 4026, 7359, 12727, 21021, 33397, 51324, 76636, 111588, 158916, 221901, 304437, 411103, 547239, 719026, 933570, 1198990, 1524510, 1920555, 2398851, 2972529, 3656233, 4466232, 5420536, 6539016, 7843528, 9358041, 11108769, 13124307

Offset

0,3

Comments

After 0, second bisection of A129548.

This sequence is also the total number of squares of all sizes in i X i subsquares in an n X n grid, whereas A000330 simply gives the number of all sizes of squares in an n X n grid. See illustrations.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Luce ETIENNE, Illustration of initial terms.

Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*(n-i)^2.

a(n) = Sum_{i=0..n} A000290(n-i)*A000330(i+1).

G.f.: x*(1 + x)^2 / (1 - x)^7. - Colin Barker, Nov 08 2015

a(n) = (A000539(n+1) - A000217(n+1))/30. - Yasser Arath Chavez Reyes, Feb 24 2024

Sum_{n>=1} 1/a(n) = 48*sqrt(5)*Pi*tanh(sqrt(5)*Pi/2)/7 - 2300/49. - Amiram Eldar, Jul 18 2026

Example

a(0) = 0; a(1) = 1*1; a(2) = 4*1+1*5 = 9; a(3) = 9*1+4*5+1*14 = 43.

Mathematica

A259181[n_] := n*(n + 1)*(n + 2)*(n + 3)*(2*n*(n + 3) + 7)/360;
Array[A259181, 40, 0] (* Paolo Xausa, Jul 13 2026 *)

Prog

(PARI) vector(100, n, n--; n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360) \\ Altug Alkan, Nov 08 2015
(PARI) concat(0, Vec(-x*(x+1)^2 / (x-1)^7 + O(x^100))) \\ Colin Barker, Nov 08 2015

Crossrefs

Cf. A000217, A000290, A000330, A000539, A129548.

Cf. A060060 ((1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*i^2).

Sequence in context: A332373 A181945 A244869 * A330088 A281773 A220676

Adjacent sequences: A259178 A259179 A259180 * A259182 A259183 A259184

Keyword

nonn,easy,changed

Author

Luce ETIENNE, Nov 08 2015

Status

approved