A302335 Constant coefficient of the quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1.

0, 1, 4, 26, 164, 1046, 6672, 42790, 275888, 1787624, 11634704

Offset

1,3

Comments

a(n) is the sum of the areas of minimal bounding rectangles of (fixed, self-avoiding) 2n-cycles in a grid. - Andrey Zabolotskiy, Feb 09 2022

Links

Table of n, a(n) for n=1..11.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Grid Graph

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 constant coefficients give a(n), so the first few are 0, 1, 4, 26, 164, .... - Eric W. Weisstein, Apr 05 2018

Crossrefs

Cf. A302336 (linear coefficients).

Cf. A002931 (quadratic coefficients).

Sequence in context: A092167 A124544 A145840 * A244787 A220305 A210911

Adjacent sequences: A302332 A302333 A302334 * A302336 A302337 A302338

Keyword

nonn,more

Author

Eric W. Weisstein, Apr 05 2018

Status

approved