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A008731 Molien series for 3-dimensional group [2,n ] = *22n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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]

LINKS

Table of n, a(n) for n=0..62.

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).

FORMULA

a(n) = (1/48) {2n^2 + 14n + 27 + (6n+21)(-1)^n - 16[n is 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

EXAMPLE

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 + ...

MAPLE

1/(1-x^2)^2/(1-x^3)

MATHEMATICA

nn=54; CoefficientList[Series[1/(1-x^2)^2/(1-x^3), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 13 2013 *)

a[ n_] := Quotient[ (2 n^2 + If[ OddQ[n], 8 n + 6, 20 n + 48]), 48]; (* 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 *)

PROG

(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 */

CROSSREFS

First differences of A008763.

Sequence in context: A029138 A161051 A161255 * A114209 A132091 A262090

Adjacent sequences:  A008728 A008729 A008730 * A008732 A008733 A008734

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 22 11:30 EST 2018. Contains 299452 sequences. (Running on oeis4.)