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 A231430 Number of ternary sequences which contain 000. 4
 0, 0, 0, 1, 5, 21, 81, 295, 1037, 3555, 11961, 39667, 130049, 422403, 1361385, 4359115, 13880129, 43984227, 138795849, 436367131, 1367434577, 4272615603, 13315096089, 41397076939, 128429930465, 397665266595, 1229127726825, 3792875384251, 11686625364785 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Recurrence formula given below, a(n) = 3*a(n-1) + 2* (3^(n-4) - a(n-4)) based on following recursive construction: To a string of length (n-1) containing 000 add any of {0,1,2}.  To a string of length (n-4) NOT containing 000, add 1000 or 2000. These two operations result in the two terms of the formula. LINKS FORMULA a(n) = 3*a(n-1) + 2* (3^(n-4) - a(n-4)). G.f.: x^3/(1 - 5*x + 4*x^2 + 4*x^3 +6*x^4). - Geoffrey Critzer, Jan 14 2014 EXAMPLE For n = 3, the only string is 000. For n = 4, the 5 strings are: 0000,0001,0002,1000,2000. For n = 5, there are: 1 with 5 0's, 12 with 4 0's, and 8 with just 3; total 21. MATHEMATICA t = {0, 0, 0, 1}; Do[AppendTo[t, 3 t[[-1]] + 2*(3^(n - 4) - t[[-4]])], {n, 4, 30}]; t (* T. D. Noe, Nov 11 2013 *) (* or *) nn=28; r=Solve[{s==2x s+2x a+2x b+1, a==x s, b==x a, c==3x c+x b}, {s, a, b, c}]; CoefficientList[Series[c/.r, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 14 2014 *) CROSSREFS Cf. A119826 (without 000), A119827 (exactly one 000). Cf. A186244 (with 00). Sequence in context: A029870 A269915 A273393 * A269916 A273205 A269917 Adjacent sequences:  A231427 A231428 A231429 * A231431 A231432 A231433 KEYWORD nonn AUTHOR Toby Gottfried, Nov 09 2013 STATUS approved

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Last modified May 12 11:51 EDT 2021. Contains 343821 sequences. (Running on oeis4.)