A063494 a(n) = (2*n - 1)*(7*n^2 - 7*n + 3)/3.

1, 17, 75, 203, 429, 781, 1287, 1975, 2873, 4009, 5411, 7107, 9125, 11493, 14239, 17391, 20977, 25025, 29563, 34619, 40221, 46397, 53175, 60583, 68649, 77401, 86867, 97075, 108053, 119829, 132431, 145887, 160225, 175473, 191659, 208811

Offset

1,2

Comments

Interpret A176271 as an infinite square array read by antidiagonals, with rows 1,5,11,19,...; 3,9,17,27,... and so on. The sum of the terms in the n X n upper submatrix are s(n) = 1, 18, 93, 296, ... = n^2*(7*n^2-1)/6, and a(n) = s(n) - s(n-1) are the first differences. - J. M. Bergot, Jun 27 2013

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).

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

Formula

G.f.: x*(1+x)*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012

E.g.f.: (-3 + 6*x + 21*x^2 + 14*x^3)*exp(x)/3 + 1. - G. C. Greubel, Dec 01 2017

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, May 11 2023

Mathematica

Table[(2*n - 1)*(7*n^2 - 7*n + 3)/3, {n, 1, 30}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 17, 75, 203}, 30] (* G. C. Greubel, Dec 01 2017 *)

Prog

(PARI) { for (n=1, 1000, write("b063494.txt", n, " ", (2*n - 1)*(7*n^2 - 7*n + 3)/3) ) } \\ Harry J. Smith, Aug 23 2009
(Magma) [(2*n - 1)*(7*n^2 - 7*n + 3)/3: n in [1..30]]; // G. C. Greubel, Dec 01 2017
(PARI) x='x+O('x^30); Vec(serlaplace((-3+6*x+21*x^2+14*x^3)*exp(x)/3 + 1)) \\ G. C. Greubel, Dec 01 2017

Crossrefs

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Sequence in context: A097223 A296113 A231779 * A146594 A202138 A357740

Adjacent sequences: A063491 A063492 A063493 * A063495 A063496 A063497

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 01 2001

Status

approved