A245487 Number of compositions of n into parts 3,4 where both parts are always present.

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 3, 0, 4, 6, 4, 5, 10, 10, 11, 15, 20, 22, 27, 35, 43, 49, 63, 79, 92, 112, 144, 171, 204, 257, 316, 375, 462, 573, 692, 838, 1035, 1265, 1532, 1873, 2300, 2798, 3406, 4173, 5099, 6204, 7580, 9273, 11303, 13784, 16855, 20576

Offset: 0

Author

David Neil McGrath, Jul 23 2014

Keywords

• nonn
• easy

Examples

			a(16)=5, the compositions being 43333, 34333, 33433, 33343, 33334.
		

Links

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

Crossrefs

Cf. A245332.

Programs

Mathematica

CoefficientList[Series[x^7 (x^3 + 2 x^2 + 2 x + 2)/((x - 1) (x + 1) (x^2 + 1) (x^2 + x + 1) * (x^4 + x^3 - 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Jul 25 2014 *)

PARI

a=[0,0,0,0,0,0,2,0,0,3]; b=[1,0,1,0,1,1,0,0,2,0,0,1]; k=1; for(n=11, 100, a=concat(a, a[n-3]+a[n-4]+b[k]); if(k==#b, k=1, k++)); a \\ Colin Barker, Jul 24 2014

Formula

a(n) = a(n-3)+a(n-4)+b(n) where b(n) is the 12-cycle (1,0,1,0,1,1,0,0,2,0,0,1) starting with initial value b(11)=1 and b(n)=b(n-12) e.g. b(23)=b(11). The initial values for a(n) are a(7)=2,a(8)=0,a(9)=0,a(10)=3.

G.f.: x^7*(x^3+2*x^2+2*x+2) / ((x-1)*(x+1)*(x^2+1)*(x^2+x+1)*(x^4+x^3-1)). - Colin Barker, Jul 24 2014

Extensions

More terms from Colin Barker, Jul 24 2014