A296361 Number of monohedral disk tilings of type C^t_{3,n}.

2, 62, 116, 200, 318, 476, 682, 946, 1272, 1674, 2152, 2724, 3394, 4176, 5078, 6110, 7284, 8614, 10108, 11784, 13646, 15716, 18002, 20522, 23288, 26314, 29616, 33212, 37114, 41344, 45910, 50838, 56140, 61838, 67948, 74488, 81478, 88940, 96890, 105354, 114344

Offset: 1

Author

N. J. A. Sloane, Dec 15 2017

Keywords

• nonn

Links

• Lars Blomberg, Table of n, a(n) for n = 1..1000
• Joel Anthony Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv:1512.03794v2 [math.MG], 2015-2016.

Crossrefs

Cf. A241926, A296359, A296360, A296362.

Programs

Mathematica

U[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[(n + k)/#, n/#]/(n + k) &];
a[1] = 2; a[n_] := 2*Sum[U[i, n*(6 - i)], {i, 0, 6}];
Array[a, 50] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

PARI

\\ here U is A241926
U(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial((n+k)/d, n/d)/(n+k))}
a(n)={2*if(n<2, n==1, sum(i=0, 6, U(i,n*(6-i))))} \\ Andrew Howroyd, Jan 09 2018

Formula

Conjectures from Colin Barker, Jan 09 2018: (Start)

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)).

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.

(End)

a(n) = 2*Sum_{i=0..6} A241926(i, n*(6-i)) for n > 1. - Andrew Howroyd, Jan 09 2018

Extensions

Terms a(6) and beyond from Lars Blomberg, Jan 09 2018