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

0,3

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

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

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

Prog

(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

Crossrefs

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

Sequence in context: A053530 A215575 A266049 * A081555 A301341 A063042

Adjacent sequences: A132099 A132100 A132101 * A132103 A132104 A132105

Keyword

nonn

Author

Keith F. Lynch, Oct 31 2007

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

Status

approved