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A255116
Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.
5
1, 3, 10, 33, 108, 354, 1161, 3807, 12483, 40932, 134217, 440100, 1443096, 4731939, 15516117, 50877639, 166828734, 547034553, 1793736576, 5881695930, 19286191449, 63239784075, 207364440015, 679951894392, 2229575035401, 7310818426248, 23972310961920
OFFSET
0,2
LINKS
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 10.
FORMULA
a(n+3) = 3*a(n+2) + 3*a(n) with n>1, a(0) = 1, a(1) = 3, a(2) = 10.
G.f.: -(x^2+1) / (3*x^3+3*x-1). - Colin Barker, Feb 15 2015
a(n) = A089978(n) + A089978(n-2). - R. J. Mathar, Aug 04 2019
MATHEMATICA
RecurrenceTable[{a[0] == 1, a[1] == 3, a[2]== 10, a[n] == 3 a[n - 1] + 3 a[n - 3]}, a[n], {n, 0, 25}]
LinearRecurrence[{3, 0, 3}, {1, 3, 10}, 30] (* Harvey P. Dale, Feb 20 2023 *)
PROG
(PARI) Vec(-(x^2+1)/(3*x^3+3*x-1) + O(x^100)) \\ Colin Barker, Feb 15 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Milan Janjic, Feb 14 2015
STATUS
approved