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.

2, 62, 2, 116, 1532, 2, 200, 6402, 50830, 2, 318, 19884, 446930, 1855110, 2, 476, 51128, 2460462, 34121322, 71292624, 2, 682, 115188, 10106370, 332112068, 2741227176, 2833906726, 2, 946, 235180, 33905948, 2177193500, 47162138964

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..33.

Joel Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv preprint arXiv:1512.03794v2 [math.MG], 2016.

Formula

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].

Example

Array begins:
.    2       2         2            2              2                 2
.   62    1532     50830      1855110       71292624        2833906726
.  116    6402    446930     34121322     2741227176      227759341712
.  200   19884   2460462    332112068    47162138964     6926365932512
.  318   51128  10106370   2177193500   493416845604   115646287581042
.  476  115188  33905948  10874491594  3668999040616  1280224897307324

Mathematica

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)}]);
(* Array: *)
Grid[Table[a[k, n], {k, 6}, {n, 6}]]
(* Or array antidiagonals flattened: *)
Flatten[Table[a[k - n + 1, n], {k, 7}, {n, k}]]

Crossrefs

Cf. A047996.

Sequence in context: A289644 A347606 A094478 * A106133 A064738 A067825

Adjacent sequences: A273087 A273088 A273089 * A273091 A273092 A273093

Keyword

nonn,tabl

Author

L. Edson Jeffery, May 14 2016

Status

approved