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A145392

Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated by a multiple of Pi/2 to give the other.

1, 2, 2, 4, 4, 6, 4, 8, 7, 10, 6, 14, 8, 12, 12, 16, 10, 20, 10, 22, 16, 18, 12, 30, 17, 22, 20, 28, 16, 36, 16, 32, 24, 28, 24, 46, 20, 30, 28, 46, 22, 48, 22, 42, 40, 36, 24, 62, 29, 48, 36, 50, 28, 60, 36, 60, 40, 46, 30, 84, 32, 48, 52, 64, 44, 72, 34, 64, 48, 72 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`From Andrey Zabolotskiy, Mar 12 2018: (Start)` `The parent lattice of the sublattices under consideration has Patterson symmetry group p4, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145393 (p4mm), A145394 (p6), A003051 (p6mm).` `If we count sublattices related by parent-lattice-preserving reflection as equivalent, we get A145393 instead of this sequence. If we count sublattices related by rotation of the sublattice only (but not parent lattice; equivalently, sublattices related by rotation by angle which is not a multiple of Pi/2; see illustration in links) as equivalent, we get A054345. If we count sublattices related by any rotation or reflection as equivalent, we get A054346.` `Rutherford says at p. 161 that a(n) != A054345(n) only when A002654(n) > 1, but actually these two sequences differ at other terms, too, for example, at n = 15 (see illustration). (End)`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 1..16384` `John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2; beware the typo in a(13).]` `Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)` `Index entries for sequences related to sublattices` `Index entries for sequences related to square lattice`
	FORMULA	`a(n) = (A000203(n) + A002654(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009` `a(n) = Sum_{ m: m^2\|n } A000089(n/m^2) + A157224(n/m^2) = A002654(n) + Sum_{ m: m^2\|n } A157224(n/m^2). - Andrey Zabolotskiy, May 07 2018` `a(n) = Sum_{ d\|n } A004525(d). - Andrey Zabolotskiy, Aug 29 2019`
	PROG	`(PARI)` `A002654(n) = sumdiv(n, d, (d%4==1) - (d%4==3));` `A145392(n) = ((sigma(n) + A002654(n))/2); \\ Antti Karttunen, Nov 23 2017`
	CROSSREFS	`Cf. A000203, A002654, A069734, A145391, A145393, A145394, A003051, A054345, A054346, A000089, A157224, A004525.` `Sequence in context: A075857 A023817 A300754 * A034974 A048275 A211508` `Adjacent sequences: A145389 A145390 A145391 * A145393 A145394 A145395`
	KEYWORD	`nonn`
	AUTHOR	`N. J. A. Sloane, Feb 23 2009`
	EXTENSIONS	`New name from Andrey Zabolotskiy, Mar 12 2018`
	STATUS	`approved`