A132102 Number of distinct Tsuro tiles which are n-gonal in shape and have 2 points per side.

1, 1, 3, 7, 35, 193, 1799, 19311, 254143, 3828921, 65486307, 1249937335, 26353147811, 608142583137, 15247011443103, 412685556939751, 11993674252049647, 372509404162520641, 12313505313357313047, 431620764875678503143, 15991549339008732109899

Offset: 0

Author

Keith F. Lynch, Oct 31 2007

Keywords

• nonn

Comments

Turning over is not allowed.

See A132100 for definition and comments.

Even and odd terms can be computed with the help of Burnside Lemma and recursive sequences. - Lionel RAVEL, Sep 18 2013

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A001147, A007769, A054499, A132100, A132101, A132103, A132104, A132105.

Programs

Maple

with(numtheory): a:=(p,q)->piecewise(p mod 2 = 1, p^q*doublefactorial(2*q - 1), sum(p^j*binomial(2*q, 2*j)*doublefactorial(2*j - 1), j = 0 .. q));
A132102 := n->add(phi(p)*a(p,n/p),p in divisors(n))/n;
[seq(A132102(n),n=1..20)]; # Laurent Tournier, Jul 09 2014

PARI

a(n)={if(n<1, n==0, sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(j=0, m, (d%2==0 || m-j==0) * binomial(2*m, 2*j) * d^j * (2*j)! / (j!*2^j) ))/n)} \\ Andrew Howroyd, Jan 26 2020

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*alpha(d, n/d), where phi = Euler's totient function,

alpha(p,q) = Sum_{j=0..q} p^j * binomial(2q, 2j) * (2j-1)!! if p even,

= p^q * (2q-1)!! if p odd. (cf. also A132100) - Laurent Tournier, Jul 09 2014

Extensions

More terms from Lionel RAVEL, Sep 18 2013

a(9) and a(10) corrected, and addition of more terms using formula given above by Laurent Tournier, Jul 09 2014