A162147 a(n) = n*(n+1)*(5*n + 4)/6.

0, 3, 14, 38, 80, 145, 238, 364, 528, 735, 990, 1298, 1664, 2093, 2590, 3160, 3808, 4539, 5358, 6270, 7280, 8393, 9614, 10948, 12400, 13975, 15678, 17514, 19488, 21605, 23870, 26288, 28864, 31603, 34510, 37590, 40848, 44289, 47918, 51740, 55760

Offset

0,2

Comments

Partial sums of A005475.

Suppose we extend the triangle in A215631 to a symmetric array by reflection about the main diagonal. The array is defined by m(i,j) = i^2 + i*j + j^2: 3, 7, 13, ...; 7, 12, 19, ...; 13, 19, 27, .... Then a(n) is the sum of the n-th antidiagonal. Examples: 3, 7 + 7, 13 + 12 + 13, 21 + 19 + 19 + 21, etc. - J. M. Bergot, Jun 25 2013

Binomial transform of [0,3,8,5,0,0,0,...]. - Alois P. Heinz, Mar 10 2015

Links

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

From R. J. Mathar, Jun 27 2009: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)

a(n) = A033994(n) + A000217(n).

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

a(n) = A035005(n+1)/4. - Johannes W. Meijer, Feb 04 2010

a(n) = Sum_{i=0..n} i*(n + 1 + i). - Bruno Berselli, Mar 17 2016

E.g.f.: x*(18 + 24*x + 5*x^2)*exp(x)/6. - G. C. Greubel, Apr 01 2021

Example

For n=4, a(4) = 0*(5+0) + 1*(5+1) + 2*(5+2) + 3*(5+3) + 4*(5+4) = 80. - Bruno Berselli, Mar 17 2016

Maple

A162147:= n-> n*(n+1)*(5*n+4)/6; seq(A162147(n), n=0..40); # G. C. Greubel, Apr 01 2021

Mathematica

Table[(n(n+1)(5n+4))/6, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 3, 14, 38}, 50] (* Harvey P. Dale, May 04 2013 *)

Prog

(PARI) a(n)=n*(n+1)*(5*n+4)/6 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [n*(n+1)*(5*n+4)/6: n in [0..40]]; // G. C. Greubel, Apr 01 2021
(Sage) [n*(n+1)*(5*n+4)/6 for n in (0..40)] # G. C. Greubel, Apr 01 2021

Crossrefs

Cf. A002412, A002413, A006331, A016061, A033994.

Sequence in context: A068044 A141129 A143941 * A319791 A027444 A000263

Adjacent sequences: A162144 A162145 A162146 * A162148 A162149 A162150

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Extensions

Definition rephrased by R. J. Mathar, Jun 27 2009

Status

approved