

A219766


Number of nonsquare simple perfect squared rectangles of order n up to symmetry.


3



0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633765, 63301415, 228130900
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OFFSET

1,9


COMMENTS

A squared rectangle (which may be a square, but not in this particular sequence) is a rectangle dissected into a finite number, two or more, of integer sized squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares.


REFERENCES

See A006983 and A217156 for references.


LINKS

Table of n, a(n) for n=1..24.
Stuart E Anderson Simple Perfect Squared Rectangles [Nonsquare rectangles only]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée AixMarseille II, 1999, p. 24.
See A006983 and A217156 for further links.


FORMULA

a(n) = A002839(n)  A006983(n).
In 'A Census of Planar Maps', William Tutte gave an asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order):
a(n) ~ ((n^(5/2))*(4^n))/(2^5*sqrt(Pi)).


MATHEMATICA

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
A002839 = A@002839;
A006983 = A@006983;
a[n_] := A002839[[n]]  A006983[[n]];
a /@ Range[24] (* JeanFrançois Alcover, Jan 13 2020 *)


CROSSREFS

Cf. A002839, A006983, A002962, A002881, A181735.
Cf. A217153, A217154, A217156.
Sequence in context: A321626 A126171 A228396 * A002839 A109194 A014334
Adjacent sequences: A219763 A219764 A219765 * A219767 A219768 A219769


KEYWORD

nonn,hard,more


AUTHOR

Stuart E Anderson, Nov 27 2012


EXTENSIONS

a(9)a(24) enumerated Gambini 1999, confirmed by Stuart E Anderson Dec 07 2012


STATUS

approved



