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

2,3

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

Table of n, a(n) for n=2..22.

Eric Weisstein's World of Mathematics, Chordless Cycle

Eric Weisstein's World of Mathematics, White Bishop Graph

Formula

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

Prog

(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

Crossrefs

Cf. A070968.

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

Sequence in context: A074361 A126187 A215420 * A294251 A294392 A358284

Adjacent sequences: A303539 A303540 A303541 * A303543 A303544 A303545

Keyword

nonn

Author

Eric W. Weisstein, Apr 25 2018

Extensions

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

Status

approved