A357442 Consider a clock face with 2*n "hours" marked around the dial; a(n) = number of ways to match the even hours to the odd hours, modulo rotations and reflections.

1, 1, 3, 5, 17, 53, 260, 1466, 10915, 93196, 917898, 10015299, 119914982, 1557364352, 21797494987, 326930305166, 5230756117008, 88922108947567, 1600594738591550, 30411281088326498, 608225534389576956, 12772735698577492558

Offset: 1

Author

N. J. A. Sloane, Nov 06 2022, based on an email from Barry Cipra, Oct 26 2022

Keywords

• nonn

Links

• Barry Cipra, Illustration for a(5) = 17
• N. J. A. Sloane, Sketch illustrating a(1) = a(2) = 1, a(3) = 3
• Philip Todd, Theorem Discovery Amongst Cyclic Polygons, arXiv:2401.13002 [cs.CG], 2024.

Crossrefs

Cf. A000031, A000699, A007769, A059375.

Programs

PARI

{ a357442(n) = ( sumdiv(n,d,(n\d)!*d^(n\d)*eulerphi(d)) + n*sum(k=0,n\2,n!\k!\2^k\(n-2*k)!) + if(n%2, n*((n-1)\2)!*2^((n-1)\2) + sumdiv(n,d, eulerphi(d)*sum(k=0,n\d\2,(n\d)! \ (2*k+1)! \ ((n\d-1)\2-k)! * (d/2)^((n\d-1)\2-k) ))) )\n\4; } \\ Max Alekseyev, Nov 10 2022

Formula

See PARI code for the formula. - Max Alekseyev, Nov 10 2022

Extensions

Terms a(7) onward from Max Alekseyev, Nov 10 2022