A181137 The number of ways to color n balls in a row with 3 colors with no color runs having lengths greater than 3.

1, 3, 9, 27, 78, 228, 666, 1944, 5676, 16572, 48384, 141264, 412440, 1204176, 3515760, 10264752, 29969376, 87499776, 255467808, 745873920, 2177683008, 6358049472, 18563212800, 54197890560, 158238305664, 461998818048, 1348870028544, 3938214304512

Offset: 0

Author

William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 06 2010

Keywords

• nonn
• easy

Comments

This sequence is a special case of the general problem for coloring n balls in a row with p colors where each color has a given maximum run-length. In this example, the bounds are uniformly 3. It can be phrased in terms of tossing a p-faced die n times, requiring each face to have no runs longer than b.

Generating function and recurrence for given p and uniform bound b are known. a(n+b) = (p-1)(a(n)+ ... + a(n+b-1)), using b initial values a(1)=p, a(2)=p^2, ..., a(b)=p^(b) The g.f. is p*G/(1-(p-1)*G) where G = t + t^2 + ... + t^b.

Examples

			For p=3 and b=3, a(4)=78. The colorings are: 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1311, 1312, 1313, 1321, 1322, 1323, 1331, 1332, 1333, 2111, 2112, 2113, 2121, 2122, 2123, 2131, 2132, 2133, 2211, 2212, 2213, 2221, 2223, 2231, 2232, 2233, 2311, 2312, 2313, 2321, 2322, 2323, 2331, 2332, 2333, 3111, 3112, 3113, 3121, 3122, 3123, 3131, 3132, 3133, 3211, 3212, 3213, 3221, 3222, 3223, 3231, 3232, 3233, 3311, 3312, 3313, 3321, 3322, 3323, 3331, 3332.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Li Guo and William Sit, Enumeration and Generating Functions for Differential Rota-Baxter Words, Math. Comput. Sci. 4 (2010), 339-358.
• Index entries for linear recurrences with constant coefficients, signature (2,2,2).

Crossrefs

A135492 is sequence[2, {2, 4, 6, 8}, n-4], for colorings of n balls in a row with p=2 colors so no color has run length more than 4. A135491 coloring of 2 balls in a row with p=2 colors so no color has run length more than 3. In general 2 colorings are like coin tossing. The example here is 3 colorings (tossing 3-sided dice).

Column 3 in A265624.

Programs

Mathematica

(* next[p,z] computes the next member in a sequence and
next[p,z] = a(n+b)= (p-1)( c(b)+ ... + c(n+b-1)) where z is the preceding b items on the sequence starting with a(n) where b is the uniform bound on runs.
The function sequence[p,z,n] computes the next n terms. *) next[p_,z_]:=(p-1) Apply[Plus,z] sequence[p_,z_,n_]:=Module[{y=z,seq=z, m=n, b=Length[z]}, While[m>0, seq = Join[seq,{next[p,y]}]; y = Take[seq, -b]; m-- ]; seq] (* sequence[3,{3,9,27},10] computes the next 10 terms after 3,9,27. *)
LinearRecurrence[{2,2,2},{1,3,9,27},30] (* Harvey P. Dale, Dec 01 2017 *)

PARI

Vec((1 + x)*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3) + O(x^30)) \\ Colin Barker, Jun 28 2019

Formula

G.f.: 1+3t(t^2+t+1)/(1 - 2t(t^2+t+1)).

a(n+3) = 2(a(n)+a(n+1)+a(n+2)), a(0)=1, a(1)=3, a(2)=9, a(3)=27.

a(n) = 3*A119826(n-1). - R. J. Mathar, Dec 10 2015

G.f.: (1 + x)*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3). - Colin Barker, Jun 28 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 10 2015