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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A181005 A289644 A094478 * A064738 A067825 A239417

Adjacent sequences:  A273087 A273088 A273089 * A273091 A273092 A273093

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, May 14 2016

STATUS

approved

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Last modified November 27 03:27 EST 2020. Contains 338677 sequences. (Running on oeis4.)