A132102 - OEIS

%I #19 Jan 26 2020 18:20:33

%S 1,1,3,7,35,193,1799,19311,254143,3828921,65486307,1249937335,

%T 26353147811,608142583137,15247011443103,412685556939751,

%U 11993674252049647,372509404162520641,12313505313357313047,431620764875678503143,15991549339008732109899

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

%C Turning over is not allowed.

%C See A132100 for definition and comments.

%C Even and odd terms can be computed with the help of Burnside Lemma and recursive sequences. - _Lionel RAVEL_, Sep 18 2013

%H Andrew Howroyd, <a href="/A132102/b132102.txt">Table of n, a(n) for n = 0..200</a>

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

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

%F = p^q * (2q-1)!! if p odd. (cf. also A132100) - _Laurent Tournier_, Jul 09 2014

%p 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));

%p A132102 := n->add(phi(p)*a(p,n/p),p in divisors(n))/n;

%p [seq(A132102(n),n=1..20)]; # _Laurent Tournier_, Jul 09 2014

%o (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

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

%K nonn

%O 0,3

%A _Keith F. Lynch_, Oct 31 2007

%E More terms from _Lionel RAVEL_, Sep 18 2013

%E a(9) and a(10) corrected, and addition of more terms using formula given above by _Laurent Tournier_, Jul 09 2014