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

0,2

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

Table of n, a(n) for n=0..35.

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).

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)

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 *)

Prog

(Magma) [(n+1)*(2*n+1)^2 : n in [0..30]]; // Wesley Ivan Hurt, Sep 29 2014
(PARI) a(n) = (n+1)*(2*n+1)^2; \\ Altug Alkan, Dec 20 2017

Crossrefs

Cf. A000217, A006254.

Sequence in context: A022145 A284659 A143666 * A285918 A262402 A296363

Adjacent sequences: A139754 A139755 A139756 * A139758 A139759 A139760

Keyword

easy,nonn

Author

Odimar Fabeny, May 19 2008

Extensions

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

Status

approved