A139757 a(n) = (n+1)*(2n+1)^2.

1, 18, 75, 196, 405, 726, 1183, 1800, 2601, 3610, 4851, 6348, 8125, 10206, 12615, 15376, 18513, 22050, 26011, 30420, 35301, 40678, 46575, 53016, 60025, 67626, 75843, 84700, 94221, 104430, 115351, 127008, 139425, 152626, 166635, 181476

Offset: 0

Author

Odimar Fabeny, May 19 2008

Keywords

• easy
• nonn

Comments

Also the detour index of the (n+1)-antiprism graph and (n+1)-cocktail party graphs for n>=2. - Eric W. Weisstein, Jul 15 2011 and Dec 20 2017

Links

• Eric Weisstein's World of Mathematics, Antiprism Graph
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Eric Weisstein's World of Mathematics, Detour Index
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A006254.

Programs

Magma

[(n+1)*(2*n+1)^2 : n in [0..30]]; // Wesley Ivan Hurt, Sep 29 2014

Maple

A139757:=n->(n+1)*(2*n+1)^2: seq(A139757(n), n=0..30); # Wesley Ivan Hurt, Sep 29 2014

Mathematica

Table[(n + 1) (2 n + 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Sep 29 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {18, 75, 196, 405}, {0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)
CoefficientList[Series[(1 + 14 x + 9 x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 20 2017 *)

PARI

a(n) = (n+1)*(2*n+1)^2; \\ Altug Alkan, Dec 20 2017

Formula

a(n) = (2n+1) * A000217(2n+1).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4); G.f.: (1+14*x+9*x^2)/(x-1)^4. - R. J. Mathar, Sep 19 2010

a(n) = Sum_{i=1..2n-1} (n^2 + n*i - i). - Wesley Ivan Hurt, Sep 29 2014

From Amiram Eldar, Jun 28 2020: (Start)

Sum_{n>=0} 1/a(n) = Pi^2/4 - log(4).

Sum_{n>=0} (-1)^n/a(n) = 2*G + log(2) - Pi/2, where G is the Catalan constant (A006752). (End)

Extensions

Missing a(0) inserted by R. J. Mathar, Sep 19 2010