A008731 Molien series for 3-dimensional group [2, n] = *22n.

1, 0, 2, 1, 3, 2, 5, 3, 7, 5, 9, 7, 12, 9, 15, 12, 18, 15, 22, 18, 26, 22, 30, 26, 35, 30, 40, 35, 45, 40, 51, 45, 57, 51, 63, 57, 70, 63, 77, 70, 84, 77, 92, 84, 100, 92, 108, 100, 117, 108, 126, 117, 135, 126, 145, 135, 155, 145, 165, 155, 176, 165, 187

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n+4) is the number of solutions to the equation X + Y + Z = n such that X < Z, Y < Z, and X + Y >= Z. - Geoffrey Critzer, Jul 13 2013

Number of partitions of n into two sorts of 2, and one sort of 3. - Joerg Arndt, Jul 14 2013

Examples

			a(4) = 3 because we have:
1 + 3 + 4 = 2 + 2 + 4 = 3 + 1 + 4. - _Geoffrey Critzer_, Jul 13 2013
G.f. = 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 5*x^6 + 3*x^7 + 7*x^8 + 5*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 222
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

Crossrefs

First differences of A008763.

Programs

GAP

a:=[1,0,2,1,3,2,5];; for n in [8..70] 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, Jul 30 2019

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Jul 30 2019

Maple

seq(coeff(series(1/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0 .. 70); # modified by G. C. Greubel, Jul 30 2019

Mathematica

CoefficientList[Series[1/(1-x^2)^2/(1-x^3),{x,0,70}],x] (* Geoffrey Critzer, Jul 13 2013 *)
a[ n_] := Quotient[ (2 n^2 + If[ OddQ[n], 8 n + 6, 20 n + 48]), 70]; (* Michael Somos, Feb 02 2015 *)
a[ n_] := Module[{m=n}, If[ n < 0, m=-7-n]; SeriesCoefficient[ 1 / ( (1 - x^2)^2 * (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Feb 02 2015 *)
LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,3,2,5},70] (* Harvey P. Dale, Feb 23 2018 *)

PARI

{a(n) = (2*n^2 + if( n%2, 8*n + 6, 20*n + 48)) \ 48}; /* Michael Somos, Feb 02 2015 */

PARI

{a(n) = if( n<0, n=-7-n); polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3)) + x * O(x^n), n)}; /* Michael Somos, Feb 02 2015 */

Sage

(1/((1-x^2)^2*(1-x^3))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

Formula

G.f.: 1/((1-x^2)^2*(1-x^3)) = 1/((1-x)^3*(1+x)^2*(1+x+x^2)).

a(n) = (1/48)*(2*n^2 + 14*n + 27 + (6*n+21)*(-1)^n - 16(n=1 mod 3)).

Euler transform of length 3 sequence [ 0, 2, 1]. - Michael Somos, Feb 02 2015

a(n) = a(-7-n) for all n in Z. - Michael Somos, Feb 02 2015

0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - a(n+4) + a(n+5) + a(n+6) - 1 for all n in Z. - Michael Somos, Feb 02 2015

a(n+3) - a(n) = 0 if n even, (n+5)/2 otherwise. - Michael Somos, Feb 02 2015

|a(n)-a(n-1)| = A154958(n). - R. J. Mathar, Aug 11 2021