A302336 Linear coefficient (in absolute value) of the quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1.

0, 2, 6, 28, 140, 740, 4056, 22904, 132344, 778832, 4652404, 28140536, 172021360, 1061153560, 6597813620, 41307119692, 260198053200, 1647958588568, 10488324116052, 67046234983840, 430300354820176, 2771678138269600, 17912347088664868, 116113406138798112

Offset

1,2

Links

Table of n, a(n) for n=1..24.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Grid Graph

Formula

a(n) = 2*A006772(n). - Andrey Zabolotskiy, Nov 09 2018

Example

Let p(k,n) be the number of 2k-cycles in the n X n grid graph for n >= k-1. p(k,n) are quadratic polynomials in n, with the first few given by:
  p(1,n) =      0,
  p(2,n) =      1 -      2*n +       n^2,
  p(3,n) =      4 -      6*n +     2*n^2,
  p(4,n) =     26 -     28*n +     7*n^2,
  p(5,n) =    164 -    140*n +    28*n^2,
  p(6,n) =   1046 -    740*n +   124*n^2,
  p(7,n) =   6672 -   4056*n +   588*n^2,
  p(8,n) =  42790 -  22904*n +  2938*n^2,
  p(9,n) = 275888 - 132344*n + 15268*n^2,
  ...
The linear coefficients give a(n), so the first few are 0, 2, 6, 28, 140, .... - Eric W. Weisstein, Apr 05 2018

Crossrefs

Cf. A302335 (constant coefficients).

Cf. A002931 (quadratic coefficients).

Cf. A006772, A302337.

Sequence in context: A307523 A065577 A227294 * A225877 A350524 A228842

Adjacent sequences: A302333 A302334 A302335 * A302337 A302338 A302339

Keyword

nonn

Author

Eric W. Weisstein, Apr 05 2018

Extensions

Terms a(12) and beyond added using data from A006772 by Andrey Zabolotskiy, Feb 10 2022

Status

approved