A255015 - OEIS

A255015

Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Michael De Vlieger, Table of n, a(n) for n = 0..57` `S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.` `Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.` `Wikipedia, Polya enumeration theorem.`
	FORMULA	`a(n) = (2n^2)^{-1} Sum_{ c divides n } Sum_{ d divides n } phi(c)phi(d)* 2^(n^2/lcm(c,d)) + (2n)^{-1} Sum_{ d divides n } phi(d)2^(n(n + d - 2 floor(d/2))/(2*d)), where phi is A000010.`
	MATHEMATICA	`a[n_] := (2 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];`
	CROSSREFS	`Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).` `Sequence in context: A219337 A259482 A332757 * A347984 A229836 A359659` `Adjacent sequences: A255012 A255013 A255014 * A255016 A255017 A255018`
	KEYWORD	`nonn`
	AUTHOR	`Stewart N. Ethier, Feb 12 2015`
	EXTENSIONS	`a(0)=1 from Alois P. Heinz, Feb 19 2015`
	STATUS	`approved`