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A303542 Number of chordless cycles in the n X n white bishop graph.

%I A303542 #25 May 27 2025 14:59:41
%S A303542 0,1,3,19,97,678,5098,52170,582342,8221455,125339157,2312227461,
%T A303542 45664819407,1056675718876,26022340062564,734233350312484,
%U A303542 21939269071805596,738213020202917421,26196923530426606903,1032994592794340235015,42808941242555092330701
%N A303542 Number of chordless cycles in the n X n white bishop graph.
%C A303542 The chordless cycles in a bishop graph are those cycles which have at most one edge on any diagonal or antidiagonal. - _Andrew Howroyd_, Apr 29 2018
%H A303542 Andrew Howroyd, <a href="/A303542/b303542.txt">Table of n, a(n) for n = 2..200</a>
%H A303542 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>.
%H A303542 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>.
%F A303542 For n > 1, a(n) = A370224(n) - A370210(n).
%o A303542 (PARI)
%o A303542 SafeMat(m)={my(d=matsize(m));((j,k)->if(j>0&&j<=d[1]&&k>0&&k<=d[2], m[j,k]))}
%o A303542 CC(sig,x)={my(v=SafeMat([;]), total=0);
%o A303542 forstep(i=#sig, 2, -1, my(t=sig[i]);
%o A303542    v=SafeMat(matrix(t, t\2, j, k, v(j,k) + x*(if(j==2&&k==1, binomial(t,2)) + v(j-2,k-1)*binomial(t-j+2,2) + v(j-1,k)*2*k*(t-j+1) + v(j,k+1)*2*k*(k+1))));
%o A303542    total+=sum(j=1,t,v(j,1)) );
%o A303542 total}
%o A303542 Bishop(n, white)=vector(n-if(white, n%2, 1-n%2), i, n-i+if(white, 1-i%2, i%2));
%o A303542 a(n) = CC(Bishop(n,1),1) \\ _Andrew Howroyd_, Apr 29 2018
%o A303542 (PARI) \\ CCGenRook, Bishop defined in A370224 (slightly faster version).
%o A303542 a(n) = subst(CCGenRook(Bishop(n,1)), y, 1) \\ _Andrew Howroyd_, May 27 2025
%Y A303542 Cf. A070968.
%Y A303542 Cf. A370210 (black bishop), A370224 (bishop).
%K A303542 nonn
%O A303542 2,3
%A A303542 _Eric W. Weisstein_, Apr 25 2018
%E A303542 a(8)-a(22) from _Andrew Howroyd_, Apr 29 2018