A277236 -id:A277236

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A277237

Number of strings of length n composed of symbols from the circular list [1,2,3,4,5,6] such that adjacent symbols in the string must be adjacent in the list. No runs of length 2 or more are allowed for symbols 1, 3 and 5.

+10
1

1, 6, 15, 39, 99, 255, 651, 1671, 4275, 10959, 28059, 71895, 184131, 471711, 1208235, 3095079, 7928019, 20308335, 52020411, 133253751, 341335395, 874350399, 2239691979, 5737093575, 14695861491, 37644235791, 96427681755, 247004624919, 632715351939, 1620733851615

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (1, 4).

FORMULA

G.f.: (1+5*x+5*x^2)/(1-x-4*x^2).

For n>=3, the recurrence is a(n) = a(n-1) + 4*a(n-2), a(1)=6, a(2)=15.

a(n) = 3*((13+3*sqrt(17))*z1^n-(13-3*sqrt(17))*z2^n)/(8*sqrt(17)), where z1=(1+sqrt(17))/2 and z2=(1-sqrt(17))/2.

EXAMPLE

For n=2 the 15 strings are: 12, 16, 21, 22, 23, 32, 34, 41, 43, 44, 54, 56, 61, 65, 66.

For n=3 the 39 strings are: 121, 122, 123, 161, 165, 166, 212, 216, 221, 222, 223, 232, 234, 321, 322, 323, 341, 343, 344, 412, 416, 432, 434, 441, 443, 444, 541, 543, 544, 561, 565, 566, 612, 616, 654, 656, 661, 665, 666.

MATHEMATICA

CoefficientList[Series[(1 + 5 x + 5 x^2)/(1 - x - 4 x^2), {x, 0, 29}], x] (* Michael De Vlieger, Oct 07 2016 *)

PROG

(PARI) Vec((1+5*x+5*x^2)/(1-x-4*x^2) + O(x^40)) \\ Michel Marcus, Oct 06 2016

CROSSREFS

Cf. A277236.

KEYWORD

nonn,easy

AUTHOR

Stefan Hollos, Oct 06 2016

STATUS

approved

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