A245332 Number of compositions of n into parts 2 and 3 with at least one 2 and one 3.

0, 0, 0, 0, 0, 2, 0, 3, 3, 4, 6, 9, 10, 16, 20, 27, 36, 49, 63, 86, 113, 150, 199, 265, 349, 465, 615, 815, 1080, 1432, 1895, 2513, 3328, 4409, 5841, 7739, 10250, 13581, 17990, 23832, 31571, 41824, 55403, 73396, 97228, 128800, 170624, 226030, 299424, 396655

Offset: 0

Author

David Neil McGrath, Jul 18 2014

Keywords

• nonn
• easy

Comments

The g.f. for compositions of n into parts r, s with both parts present is 1 + 1/(1-x^r-x^s) - 1/(1-x^r) - 1/(1-x^s). - Joerg Arndt, Jul 24 2014

Examples

			a(12) = number of rearrangements of 33222 = 5!/(3!*2!) = 10.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1, 1, 3, 2, -1, -2, -1).

Programs

Mathematica

CoefficientList[Series[x^5 (x^2 + 2 x + 2)/((x - 1) (x + 1) (x^2 + x + 1) (x^3 + x^2 - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2014 *)

PARI

a=vector(100); a[5]=2; a[7]=3; b=[1,1,0,2,0,1]; k=1; for(n=8, #a, a[n]=a[n-2]+a[n-3]+b[k]; if(k==6, k=1, k++)); a \\ Colin Barker, Jul 18 2014

PARI

x='x+O('x^66);
r=2;  s=3;
gf = 1 + 1/(1-x^r-x^s) - 1/(1-x^r) - 1/(1-x^s);
Vec(gf)
\\ Joerg Arndt, Jul 24 2014

Formula

a(n) = a(n-2)+a(n-3)+b(n) with initial terms a(5)=2,a(6)=0,a(7)=3 and b(8)=1,b(9)=1,b(10)=0,b(11)=2,b(12)=0,b(13)=1 and b(n)=b(n-6).

G.f.: x^5*(x^2+2*x+2) / ((x-1)*(x+1)*(x^2+x+1)*(x^3+x^2-1)). - Colin Barker, Jul 18 2014

Extensions

More terms from Colin Barker, Jul 18 2014