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_{i=0..n-1} A049451(i) 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
For n > 0, this sequence can be obtained by summing consecutive blocks of odd numbers where the n-th block contains the next 2n odd numbers. - Marco Zárate, Jun 15 2025
a(n) is the minimum total number of up, down, left, right moves needed to bring one token from each cell of a (2*n) X (2*n) square grid to the cell (n,n). - Maksim Kita, Dec 11 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..750
Frank 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)
a(n) = Sum_{k=0..2*n-1} (2*n*(n-1)-2*k+1). - Sean A. Irvine, Jun 19 2025
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