A370224 Number of chordless cycles in the n X n bishop graph.

0, 0, 1, 6, 34, 194, 1275, 10196, 99916, 1164684, 15912733, 250678314, 4507337214, 91329638814, 2069998845799, 52044680125128, 1443501036281272, 43878538143611192, 1455221004684341721, 52393847060853213806, 2040530475078456780538, 85617882485110184661402

Offset: 1

Author

Eric W. Weisstein, Feb 12 2024

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Bishop Graph.
• Eric Weisstein's World of Mathematics, Chordless Cycle.

Crossrefs

Cf. A370210 (black bishop), A303542 (white bishop), A370228.

Programs

PARI

\\ see also program in A303542.
CCGenRook(sig)={
 my(p=0,s=0,x='x,y='y);
 forstep(i=#sig, 1, -1, my(w=sig[i]);
   if(i<#sig, p *= y^(w-sig[i+1]));
   s += polcoef(p,1,x);
   p += x*deriv(deriv(p+y^w,y),y)/2 + 2*x*deriv(deriv(p,x),x) + 2*x*deriv(deriv(p,y),x);
   p -= x^i*polcoef(p,i,x);
 );
 s;
}
Bishop(n, white)=vector(n-if(white, n%2, 1-n%2), i, n-i+if(white, 1-i%2, i%2));
a(n) = subst(CCGenRook(Bishop(n,0)) + CCGenRook(Bishop(n,1)), y, 1) \\ Andrew Howroyd, May 27 2025

Formula

For n > 1, a(n) = A370210(n) + A303542(n).

Extensions

a(21) onwards from Andrew Howroyd, May 27 2025