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

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

0,5

Comments

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

Links

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).

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A270439 A106044 A124896 * A029136 A001479 A128861

Adjacent sequences: A008739 A008740 A008741 * A008743 A008744 A008745

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved