A180158 Number of ways are there to score a break of n points at snooker. Assuming an infinite number of reds are available, along with the usual six colors, and a break alternates red-color-red-...

1, 1, 0, 1, 2, 2, 3, 5, 7, 9, 13, 20, 28, 39, 57, 82, 116, 166, 239, 342, 488, 699, 1002, 1433, 2050, 2936, 4203, 6014, 8608, 12323, 17638, 25244, 36134, 51722, 74030, 105961, 151669, 217091, 310729, 444760, 636607, 911202, 1304240, 1866817, 2672058, 3824629

Offset: 0

Author

Rob Cummings (robc(AT)rocoto.demon.co.uk), Aug 14 2010

Keywords

• nonn
• easy

Comments

Equivalently, a(n) is the number of compositions of n with odd-indexed parts being 1 and even-indexed parts being between 2 and 7. - Andrew Howroyd, Jan 14 2020

Examples

			For n=0 there is one way to score zero points - pot nothing, potting a single red is the one way to have a break of one, a break of two is impossible (as defined here), a break of three can only be achieved with a red followed by a yellow, ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,1,1,1).

Programs

PARI

Vec((1 + x)/(1 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8) + O(x^50)) \\ Andrew Howroyd, Jan 14 2020

Formula

a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) + a(n-8) for n >= 8.

G.f.: (1 + x)/(1 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8). - Andrew Howroyd, Jan 14 2020