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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.

%I A302336 #23 Feb 10 2022 08:04:43
%S A302336 0,2,6,28,140,740,4056,22904,132344,778832,4652404,28140536,172021360,
%T A302336 1061153560,6597813620,41307119692,260198053200,1647958588568,
%U A302336 10488324116052,67046234983840,430300354820176,2771678138269600,17912347088664868,116113406138798112
%N 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.
%H A302336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A302336 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%F A302336 a(n) = 2*A006772(n). - _Andrey Zabolotskiy_, Nov 09 2018
%e A302336 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 A302336   p(1,n) =      0,
%e A302336   p(2,n) =      1 -      2*n +       n^2,
%e A302336   p(3,n) =      4 -      6*n +     2*n^2,
%e A302336   p(4,n) =     26 -     28*n +     7*n^2,
%e A302336   p(5,n) =    164 -    140*n +    28*n^2,
%e A302336   p(6,n) =   1046 -    740*n +   124*n^2,
%e A302336   p(7,n) =   6672 -   4056*n +   588*n^2,
%e A302336   p(8,n) =  42790 -  22904*n +  2938*n^2,
%e A302336   p(9,n) = 275888 - 132344*n + 15268*n^2,
%e A302336   ...
%e A302336 The linear coefficients give a(n), so the first few are 0, 2, 6, 28, 140, .... - _Eric W. Weisstein_, Apr 05 2018
%Y A302336 Cf. A302335 (constant coefficients).
%Y A302336 Cf. A002931 (quadratic coefficients).
%Y A302336 Cf. A006772, A302337.
%K A302336 nonn
%O A302336 1,2
%A A302336 _Eric W. Weisstein_, Apr 05 2018
%E A302336 Terms a(12) and beyond added using data from A006772 by _Andrey Zabolotskiy_, Feb 10 2022