%I #23 Feb 10 2022 08:04:43

%S 0,2,6,28,140,740,4056,22904,132344,778832,4652404,28140536,172021360,

%T 1061153560,6597813620,41307119692,260198053200,1647958588568,

%U 10488324116052,67046234983840,430300354820176,2771678138269600,17912347088664868,116113406138798112

%N 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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>

%F a(n) = 2*A006772(n). - _Andrey Zabolotskiy_, Nov 09 2018

%e 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:

%e p(1,n) = 0,

%e p(2,n) = 1 - 2*n + n^2,

%e p(3,n) = 4 - 6*n + 2*n^2,

%e p(4,n) = 26 - 28*n + 7*n^2,

%e p(5,n) = 164 - 140*n + 28*n^2,

%e p(6,n) = 1046 - 740*n + 124*n^2,

%e p(7,n) = 6672 - 4056*n + 588*n^2,

%e p(8,n) = 42790 - 22904*n + 2938*n^2,

%e p(9,n) = 275888 - 132344*n + 15268*n^2,

%e ...

%e The linear coefficients give a(n), so the first few are 0, 2, 6, 28, 140, .... - _Eric W. Weisstein_, Apr 05 2018

%Y Cf. A302335 (constant coefficients).

%Y Cf. A002931 (quadratic coefficients).

%Y Cf. A006772, A302337.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Apr 05 2018

%E Terms a(12) and beyond added using data from A006772 by _Andrey Zabolotskiy_, Feb 10 2022