A293820 -id:A293820 - cp's OEIS frontend

A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 1, 6, 6, 4, 1, 1, 4, 10, 13, 10, 4, 1, 1, 2, 12, 21, 21, 12, 5, 1, 1, 5, 20, 37, 41, 30, 15, 5, 1, 1, 4, 23, 51, 74, 65, 43, 19, 6, 1, 1, 7, 35, 84, 126, 131, 99, 55, 22, 6, 1, 1, 5, 38, 108, 196, 239, 216, 143, 73, 26, 7, 1, 1, 10, 56, 166, 314, 422, 428

Offset: 3

James East, Oct 16 2017

Rotations are counted only once, but reflections are considered different. For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2). Column k=3 is A008742, column k=4 is A293821, column k=5 is A293822 and column k=6 is A293823.

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

			For polygons having perimeter 7, there are: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111). Note that the quadrilaterals 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.
The triangle begins:
n=3:  1;
n=4:  0,  1;
n=5:  1,  1,  1;
n=6:  1,  2,  1,  1;
n=7:  2,  4,  3,  1,  1;
n=8:  1,  6,  6,  4,  1,  1;
n=9:  4, 10, 13, 10,  4,  1,  1;
...

Andrew Howroyd, Rows n=3..52 of triangle, flattened
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Columns: A008742 (triangles), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).
Row sums are A293820.
Same triangle with reflection allowed is A124287.

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#]&]/n - Binomial[Floor[n/2], k - 1];
Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)

T(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d,k/d))/n - binomial(floor(n/2), k-1)}
for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 21 2017

T(n,k) = (Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d))/n - binomial(floor(n/2), k-1). - Andrew Howroyd, Nov 21 2017

A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 3, 6, 13, 21, 37, 51, 84, 108, 166, 203, 294, 350, 486, 566, 759, 867, 1133, 1276, 1631, 1815, 2275, 2509, 3094, 3386, 4116, 4473, 5372, 5804, 6896, 7412, 8721, 9333, 10887, 11606, 13433, 14269, 16401, 17367, 19836, 20944, 23782, 25047, 28290, 29726, 33410, 35030, 39195, 41015

Offset: 5

James East, Oct 16 2017

nonn

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).

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

			For example, there are 6 rotation-classes of perimeter-8 pentagons: 32111, 31211, 31121, 31112, 22211, 22121.  Note that 32111 and 31112 are reflections of each other, but these are not rotationally equivalent.

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

Column k=5 of A293819.

Cf. A293820 (polygons), A293821 (quadrilaterals), A293823 (hexagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 60}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A293819 *)

G.f.: x^5*(1 + x - x^2 + 2*x^3 + 7*x^4 + 2*x^5 - 2*x^6 + x^8) / ((1 - x)^5*(1 + x)^4*(1 + x + x^2 + x^3 + x^4)) (conjectured). - Colin Barker, Nov 01 2017

A008742 Molien series for 3-dimensional group [3,3 ]+ = 332.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 2, 5, 4, 7, 5, 10, 7, 12, 10, 15, 12, 19, 15, 22, 19, 26, 22, 31, 26, 35, 31, 40, 35, 46, 40, 51, 46, 57, 51, 64, 57, 70, 64, 77, 70, 85, 77, 92, 85, 100, 92, 109, 100, 117, 109, 126, 117, 136

Offset: 0

N. J. A. Sloane

a(n) is also the number of integer-sided triangles having perimeter n + 3, modulo rotations but not reflections. - James East, Oct 16 2017

			For n = 6, there are 4 rotation-classes of perimeter-9 triangles: 441, 432, 423, 333. Note that 432 and 423 are reflections of each other, but these are not rotationally equivalent. So a(6) = 4. - _James East_, Oct 16 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
James East and Ron Niles, Integer polygons with given perimeter, arXiv/1710.11245 [math.CO], 2017.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

Cf. A005044, A293819 (k-gon triangle), A293820 (polygons), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons)

a:=[1,0,1,1,2,1,4];; for n in [8..60] do a[n]:=2*a[n-2]+a[n-3]-a[n-4] -2*a[n-5]+a[n-7]; od; a; # G. C. Greubel, Aug 03 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^6)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Aug 03 2019

CoefficientList[Series[(1+x^6)/((1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 60}], x] (* Vaclav Kotesovec, Apr 29 2014 *)

my(x='x+O('x^60)); Vec((1+x^6)/((1-x^2)*(1-x^3)*(1-x^4))) \\ G. C. Greubel, Aug 03 2019

((1+x^6)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

G.f.: (1+x^6)/((1-x^2)*(1-x^3)*(1-x^4)).
a(n) ~ 1/24*n^2. - Ralf Stephan, Apr 29 2014
a(n) = 1 - 19*n/24 - 5*n^2/24 + 4/3*floor(n/3) + (n/2+3/4)*floor(n/2) + 2/3*floor((n+1)/3). - Vaclav Kotesovec, Apr 29 2014
a(n) = floor((n^2+3*n+20)/24+(2*n+3)*(-1)^n/16). - Tani Akinari, Jun 20 2014
G.f.: (1-x^2+x^4)/((1+x+x^2)*(1+x)^2*(1-x)^3). - R. J. Mathar, Dec 18 2014

A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 12, 20, 23, 35, 38, 56, 60, 84, 88, 120, 125, 165, 170, 220, 226, 286, 292, 364, 371, 455, 462, 560, 568, 680, 688, 816, 825, 969, 978, 1140, 1150, 1330, 1340, 1540, 1551, 1771, 1782, 2024, 2036, 2300, 2312, 2600, 2613, 2925, 2938, 3276, 3290, 3654, 3668, 4060

Offset: 4

James East, Oct 16 2017

nonn

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).

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

			For example, there are 4 rotation-classes of perimeter-7 quadrilaterals: 3211, 3121, 3112, 2221. Note that 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.

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

Column k=4 of A293819.

Cf. A008742 (triangles), A293820 (polygons), A293822 (pentagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 4];
Table[a[n], {n, 4, 59}] (* Jean-François Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

Conjectures from Colin Barker, Nov 01 2017: (Start)
G.f.: x^3*(1 - x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)).
a(n) = (1/96)*(-3*(-1 + (-1)^n + 4*i*(-i)^n - 4*i*i^n) + (7 - 15*(-1)^n)*n + 3*(-1 + (-1)^n)*n^2 + 2*n^3) where i=sqrt(-1).
(End)

A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 4, 10, 21, 41, 74, 126, 196, 314, 448, 672, 912, 1302, 1692, 2334, 2937, 3927, 4828, 6292, 7579, 9679, 11466, 14378, 16808, 20748, 23968, 29198, 33388, 40188, 45564, 54264, 61047, 72033, 80484, 94164, 104587, 121429, 134134, 154672, 170016, 194810, 213200, 242880, 264730, 300002

Offset: 6

James East, Oct 16 2017

nonn

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).

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

			For example, there are 10 rotation-classes of perimeter-9 hexagons: 411111, 321111, 312111, 311211, 311121, 311112, 222111, 221211, 221121, 212121. Note that 321111 and 311112 are reflections of each other, but these are not rotationally equivalent.

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

Column k=6 of A293819.

Cf. A293820 (polygons), A293822 (pentagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 6];
Table[a[n], {n, 6, 51}] (* Jean-François Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

G.f.: x^6*(1 + x + 5*x^3 + 10*x^4 + 7*x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9) / ((1 - x)^6*(1 + x)^5*(1 - x + x^2)*(1 + x + x^2)^2) (conjectured). - Colin Barker, Nov 01 2017

A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 32, 54, 102, 180, 336, 607, 1144, 2098, 3960, 7397, 14022, 26452, 50404, 95821, 183322, 350554, 673044, 1292634, 2489502, 4797694, 9264396, 17904220, 34652962, 67125898, 130182972, 252679320, 490918440, 954505718, 1857413460, 3616951513, 7048412792, 13744169104

Offset: 3

James East, Oct 16 2017

nonn

Rotations and reversals are counted only once. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. These are row sums of A124287.
A formula is proved in Theorem 1.6 of the East and Niles article.
The same article shows that a(n) is asymptotic to 2^(n-1) / n.

			There are 10 polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon.

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.

Row sums of A124287 (k-gon triangle).

Cf. A293820 (polygons modulo rotations only).

a[n_] := DivisorSum[n, EulerPhi[n/#]*2^# &]/(2*n) + 2^Floor[(n - 3)/2] - If[Mod[n, 4] < 2, 3*2^Floor[(n - 4)/4], 2^Floor[(n + 2)/4] ];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

a(n)={sumdiv(n, d, eulerphi(n/d)*2^d)/(2*n) + 2^floor((n-3)/2) - if(n%4<2, 3*2^floor((n-4)/4), 2^floor((n+2)/4))} \\ Andrew Howroyd, Nov 21 2017

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A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

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A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.

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A008742 Molien series for 3-dimensional group [3,3 ]+ = 332.

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A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.

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A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

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A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

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