A057813 a(n) = (2*n+1)*(4*n^2+4*n+3)/3.

1, 11, 45, 119, 249, 451, 741, 1135, 1649, 2299, 3101, 4071, 5225, 6579, 8149, 9951, 12001, 14315, 16909, 19799, 23001, 26531, 30405, 34639, 39249, 44251, 49661, 55495, 61769, 68499, 75701, 83391, 91585, 100299, 109549, 119351, 129721, 140675, 152229, 164399

Offset: 0

Author

N. J. A. Sloane, Nov 07 2000

Keywords

• nonn
• easy

Comments

For n>0, 30*a(n) is the sum of the ten distinct products of 2*n-1, 2*n+1, and 2*n+3. For example, when n = 1, we sum the ten distinct products of 1, 3, and 5: 1*1*1 + 1*1*3 + 1*1*5 + 1*3*3 + 1*3*5 + 1*5*5 + 3*3*3 + 3*3*5 + 3*5*5 + 5*5*5 = 330 = 30*11 = 30*a(1). - J. M. Bergot, Apr 06 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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.

Cf. A019560, A336266.

Programs

Magma

[(2*n+1)*(4*n^2+4*n+3)/3 : n in [0..50]] // Wesley Ivan Hurt, Apr 22 2014

Maple

A057813:=n->(2*n + 1)*(4*n^2 + 4*n + 3)/3; seq(A057813(n), n=0..50); # Wesley Ivan Hurt, Apr 06 2014

Mathematica

Table[(2*n + 1)*(4*n^2 + 4*n + 3)/3, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)

PARI

P(x, y, z) = x^3 + x^2*y + x^2*z + x*y^2 + x*y*z + x*z^2 + y^3 + y^2*z + y*z^2 + z^3;
a(n) = P(2*n-1, 2*n+1, 2*n+3)/30; \\ Michel Marcus, Apr 22 2014

Formula

a(n) = 2*A050533(n) + 1. - N. J. A. Sloane, Sep 22 2004

G.f.: (1+7*x+7*x^2+x^3)/(1-x)^4. - Colin Barker, Mar 01 2012

G.f. for sequence with interpolated zeros: 1/(8*x)*sinh(8*arctanh(x)) = 1/(16*x)*( ((1 + x)/(1 - x))^4 - ((1 - x)/(1 + x))^4 ) = 1 + 11*x^2 + 45*x^4 + 119*x^6 + .... Cf. A019560. - Peter Bala, Apr 07 2017

E.g.f.: (3 + 30*x + 36*x^2 + 8*x^3)*exp(x)/3. - G. C. Greubel, Dec 01 2017

From Peter Bala, Mar 26 2024: (Start)

12*a(n) = (2*n + 1)*(a(n + 1) - a(n - 1)).

Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 3*Pi/16 - 1/2. Cf. A016754 and A336266. (End)