A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.

0, 1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, 103411, 124223, 148028, 175100, 205725, 240201, 278838, 321958, 369895, 422995, 481616, 546128, 616913, 694365, 778890

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

Keywords

• easy
• nonn

Comments

Row sums of A103219.

From Bruno Berselli, Dec 10 2010: (Start)

a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.

The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)

a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - Vaclav Kotesovec, Dec 22 2011

Also, one-half the even-indexed terms of the partial sums of A045947. - J. M. Bergot, Apr 12 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.

Programs

Maple

for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))

Mathematica

CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* Harvey P. Dale, Jan 23 2016 *)

PARI

a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1+8*x+3*x^2)/(1-x)^5.

a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013

a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - J. M. Bergot, Jun 10 2017

From Wesley Ivan Hurt, Nov 20 2021: (Start)

a(n) = Sum_{k=1..n} k * C(2*k,2).

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)

From Peter Bala, Sep 03 2023: (Start)

a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).

Second subdiagonal of A039755. (End)