OFFSET
0,2
COMMENTS
2*a(n) = (2*n)^3 is a perfect cube.
Number of edges of the product of two complete bipartite graphs, each of order 2n, K_n,n x K_n,n - Roberto E. Martinez II, Jan 07 2002
This sequence is related to A049451 by a(n) = n*A049451(n) + sum( A049451(i), i=0..n-1 ) for n>0. - Bruno Berselli, Dec 19 2013
For n>=3, also the detour index of the n-gear graph. - Eric W. Weisstein, Dec 20 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..750
F. Ellermann, Illustration of binomial transforms
Eric Weisstein's World of Mathematics, Detour Index
Eric Weisstein's World of Mathematics, Gear Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f. 4*x*(1+4*x+x^2)/ (x-1)^4. - R. J. Mathar, Feb 01 2011
From Ilya Gutkovskiy, May 25 2016: (Start)
E.g.f.: 4*x*(1 + 3*x + x^2)*exp(x).
Sum_{n>=1} 1/a(n) = zeta(3)/4. (End)
Product_{n>=1} a(n)/A280089(n) = Pi. - Daniel Suteu, Dec 26 2016
From Bruce J. Nicholson, Dec 07 2019: (Start)
a(n) = 24*A000292(n-1) + 4*n.
a(n) = 2*A007588(n) + 2*n. (End)
MATHEMATICA
4 Range[0, 40]^3 (* Harvey P. Dale, Sep 07 2016 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 32, 108}, 40] (* Harvey P. Dale, Sep 07 2016 *)
Table[4 n^3, {n, 0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)
CoefficientList[Series[(4 x (1 + 4 x + x^2))/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 20 2017 *)
PROG
(Magma) [4*n^3: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
(PARI) a(n)=4*n^3 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved