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A338426 a(n) is the number of paths a chess king can take from (0,0) to (n+1,0) touching each point in {-1,0,1} X {1,2,...,n} exactly once.

%I A338426 #17 Nov 01 2020 16:17:58
%S A338426 1,2,28,154,1206,8364,60614,432636,3104484,22235310,159360540,
%T A338426 1141875800,8182608226,58634396372,420162632840,3010793013534,
%U A338426 21574706493988,154599722419136,1107828637412194,7938463325113516,56885333141857872,407628148378295190
%N A338426 a(n) is the number of paths a chess king can take from (0,0) to (n+1,0) touching each point in {-1,0,1} X {1,2,...,n} exactly once.
%H A338426 Peter Kagey, <a href="/A338426/b338426.txt">Table of n, a(n) for n = 0..1000</a>
%H A338426 Code Golf Stack Exchange user Bubbler, <a href="https://codegolf.stackexchange.com/q/206897/53884">Counting King's Hamiltonian Paths through 3-by-N grid</a>
%F A338426 a(n) = 7*a(n-1) + 6*a(n-2) - 39*a(n-3) + 29*a(n-4) + 28*a(n-5) - 26*a(n-6) - 10*a(n-7) + 6*a(n-8) for n >= 8.
%e A338426 For n = 1, the a(1) = 2 paths are (0,0)->(1,1)->(1,0)->(1,-1)->(2,0) and (0,0)->(1,-1)->(1,0)->(1,1)->(2,0).
%e A338426 An example of one of the a(2) = 28 paths is (0,0)->(1,1)->(2,1)->(2,0)->(1,-1)->(1,0)->(2,-1)->(3,0).
%Y A338426 Cf. A140518, A158651, A193168, A272445.
%K A338426 nonn,walk
%O A338426 0,2
%A A338426 _Peter Kagey_, Oct 25 2020