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A293820 Number of integer-sided polygons having perimeter n, modulo rotations but not reflections. 7
1, 1, 3, 5, 11, 19, 43, 75, 155, 287, 567, 1053, 2063, 3859, 7455, 14089, 27083, 51463, 98855, 188697, 362675, 695155, 1338087, 2573235, 4962875, 9571195, 18496407, 35759799, 69240899, 134154259, 260235639, 505163055, 981575759, 1908619755, 3714304167, 7233118641, 14095779055 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2). These are row sums of A293819.

A formula is given in Section 6 of the East and Niles article.

The same article shows that a(n) is asymptotic to 2^n / n.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 3..200

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

FORMULA

a(n) = (Sum_{d|n} phi(n/d)*2^d)/n - 1 - 2^floor(n/2). - Andrew Howroyd, Nov 21 2017

EXAMPLE

There are 11 polygons having perimeter 7: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111).

MATHEMATICA

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];

a[n_] := Sum[T[n, k], {k, 3, n}]

Table[a[n], {n, 3, 40}] (* Jean-Fran├žois Alcover, Jun 14 2018, after Andrew Howroyd *)

PROG

(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*2^d)/n - 1 - 2^floor(n/2); \\ Andrew Howroyd, Nov 21 2017

CROSSREFS

Cf. A008742 (triangles), A293818 (reflections allowed), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).

Row sums of A293819 (k-gon triangle).

Sequence in context: A131887 A045691 A045961 * A281380 A117272 A243616

Adjacent sequences:  A293817 A293818 A293819 * A293821 A293822 A293823

KEYWORD

nonn

AUTHOR

James East, Oct 16 2017

STATUS

approved

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Last modified September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)