A255116 Number of n-length words on {0,1,2,3} in which 0 appears only in runs of length 2.

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

Colin Barker, Table of n, a(n) for n = 0..1000

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.

Index entries for linear recurrences with constant coefficients, signature (3,0,3).

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

Cf. A000930, A239333, A239340, A254657, A254600, A254664.

Sequence in context: A060557 A018920 A271943 * A006190 A020704 A289450

Adjacent sequences: A255113 A255114 A255115 * A255117 A255118 A255119

Keyword

nonn,easy

Author

Milan Janjic, Feb 14 2015

Status

approved