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

0, 1, 3, 19, 97, 678, 5098, 52170, 582342, 8221455, 125339157, 2312227461, 45664819407, 1056675718876, 26022340062564, 734233350312484, 21939269071805596, 738213020202917421, 26196923530426606903, 1032994592794340235015, 42808941242555092330701

Offset: 2

Author

Eric W. Weisstein, Apr 25 2018

Keywords

• nonn

Comments

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..200
• Eric Weisstein's World of Mathematics, Chordless Cycle.
• Eric Weisstein's World of Mathematics, White Bishop Graph.

Crossrefs

Cf. A070968.

Cf. A370210 (black bishop), A370224 (bishop).

Programs

PARI

SafeMat(m)={my(d=matsize(m));((j,k)->if(j>0&&j<=d[1]&&k>0&&k<=d[2], m[j,k]))}
CC(sig,x)={my(v=SafeMat([;]), total=0);
forstep(i=#sig, 2, -1, my(t=sig[i]);
   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))));
   total+=sum(j=1,t,v(j,1)) );
total}
Bishop(n, white)=vector(n-if(white, n%2, 1-n%2), i, n-i+if(white, 1-i%2, i%2));
a(n) = CC(Bishop(n,1),1) \\ Andrew Howroyd, Apr 29 2018

PARI

\\ CCGenRook, Bishop defined in A370224 (slightly faster version).
a(n) = subst(CCGenRook(Bishop(n,1)), y, 1) \\ Andrew Howroyd, May 27 2025

Formula

For n > 1, a(n) = A370224(n) - A370210(n).

Extensions

a(8)-a(22) from Andrew Howroyd, Apr 29 2018