cp's OEIS Frontend

A273090 Rectangular array A read by upward antidiagonals in which the entry A(n,k) in row k and column n gives the number of families of symmetric radially generated monohedral tilings of the disk (each tiling contains 2*(2*n+1)*k congruent tiles), k >= 1, n >= 1.

%I A273090 #7 May 15 2016 08:33:48
%S A273090 2,62,2,116,1532,2,200,6402,50830,2,318,19884,446930,1855110,2,476,
%T A273090 51128,2460462,34121322,71292624,2,682,115188,10106370,332112068,
%U A273090 2741227176,2833906726,2,946,235180,33905948,2177193500,47162138964
%N A273090 Rectangular array A read by upward antidiagonals in which the entry A(n,k) in row k and column n gives the number of families of symmetric radially generated monohedral tilings of the disk (each tiling contains 2*(2*n+1)*k congruent tiles), k >= 1, n >= 1.
%C A273090 Enumeration is equivalent to counting beaded necklaces of a certain class (see A047996). For details and definitions, see the arXiv preprint by Haddley and Worsley.
%H A273090 Joel Haddley, Stephen Worsley, <a href="http://arxiv.org/abs/1512.03794">Infinite families of monohedral disk tilings</a>, arXiv preprint arXiv:1512.03794v2 [math.MG], 2016.
%F A273090 A(1,n) = 2, A(k,n) = 2*Sum_{i=0..2*(2*n+1)}Sum_{d | i, d | (2*(2*n+1)-i)*k} (phi(d)/i)*binomial((2*(2*n+1)-i)*k/d+i/d-1, i/d-1), k >= 2, n >= 1 [Haddley, Worsley, Proposition 5.1].
%e A273090 Array begins:
%e A273090 .    2       2         2            2              2                 2
%e A273090 .   62    1532     50830      1855110       71292624        2833906726
%e A273090 .  116    6402    446930     34121322     2741227176      227759341712
%e A273090 .  200   19884   2460462    332112068    47162138964     6926365932512
%e A273090 .  318   51128  10106370   2177193500   493416845604   115646287581042
%e A273090 .  476  115188  33905948  10874491594  3668999040616  1280224897307324
%t A273090 a[1, n_] := 2; a[k_, n_] := 2*(1 + Sum[(1/i)*Sum[EulerPhi[d]*Binomial[(2*(2*n + 1) - i)*k/d + i/d - 1, i/d - 1], {d, Divisors[GCD[i, (2*(2*n + 1) - i)*k]]}], {i, 2*(2*n + 1)}]);
%t A273090 (* Array: *)
%t A273090 Grid[Table[a[k, n], {k, 6}, {n, 6}]]
%t A273090 (* Or array antidiagonals flattened: *)
%t A273090 Flatten[Table[a[k - n + 1, n], {k, 7}, {n, k}]]
%Y A273090 Cf. A047996.
%K A273090 nonn,tabl
%O A273090 1,1
%A A273090 _L. Edson Jeffery_, May 14 2016