A296361 - OEIS

%I #21 Jun 14 2018 05:15:58

%S 2,62,116,200,318,476,682,946,1272,1674,2152,2724,3394,4176,5078,6110,

%T 7284,8614,10108,11784,13646,15716,18002,20522,23288,26314,29616,

%U 33212,37114,41344,45910,50838,56140,61838,67948,74488,81478,88940,96890,105354,114344

%N Number of monohedral disk tilings of type C^t_{3,n}.

%H Lars Blomberg, <a href="/A296361/b296361.txt">Table of n, a(n) for n = 1..1000</a>

%H Joel Anthony Haddley, Stephen Worsley, <a href="https://arxiv.org/abs/1512.03794">Infinite families of monohedral disk tilings</a>, arXiv:1512.03794v2 [math.MG], 2015-2016.

%F Conjectures from _Colin Barker_, Jan 09 2018: (Start)

%F G.f.: 2*x*(1 + 28*x - 33*x^2 - 10*x^3 + 34*x^4 - 16*x^5 - 26*x^6 + 35*x^7 + 8*x^8 - 32*x^9 + 13*x^10) / ((1 - x)^5*(1 + x)*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - 3*a(n-6) + 2*a(n-7) + 2*a(n-8) - 3*a(n-9) + a(n-10) for n>11.

%F (End)

%F a(n) = 2*Sum_{i=0..6} A241926(i, n*(6-i)) for n > 1. - _Andrew Howroyd_, Jan 09 2018

%t U[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[(n + k)/#, n/#]/(n + k) &];

%t a[1] = 2; a[n_] := 2*Sum[U[i, n*(6 - i)], {i, 0, 6}];

%t Array[a, 50] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ *)

%o (PARI) \\ here U is A241926

%o U(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial((n+k)/d, n/d)/(n+k))}

%o a(n)={2*if(n<2, n==1, sum(i=0, 6, U(i,n*(6-i))))} \\ _Andrew Howroyd_, Jan 09 2018

%Y Cf. A241926, A296359, A296360, A296362.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Dec 15 2017

%E Terms a(6) and beyond from _Lars Blomberg_, Jan 09 2018