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

0,8

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).

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A245332.

Sequence in context: A342984 A342985 A278094 * A074734 A174956 A124182

Adjacent sequences: A245484 A245485 A245486 * A245488 A245489 A245490

Keyword

nonn,easy

Author

David Neil McGrath, Jul 23 2014

Extensions

More terms from Colin Barker, Jul 24 2014

Status

approved