A145112 Numbers of length n binary words with fewer than 4 0-digits between any pair of consecutive 1-digits.
1, 2, 4, 8, 16, 32, 63, 123, 239, 463, 895, 1728, 3334, 6430, 12398, 23902, 46077, 88821, 171213, 330029, 636157, 1226238, 2363656, 4556100, 8782172, 16928188, 32630139, 62896623, 121237147, 233692123, 450456059, 868281980, 1673667338, 3226097530, 6218502938
Offset: 0
Examples
a(6) = 63 = 2^6-1, because 100001 is the only binary word of length 6 with not less than 4 0-digits between any pair of consecutive 1-digits.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
- Index entries for linear recurrences with constant coefficients, signature (3,-2,0,0,-1,1).
Programs
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Maple
a:= n-> (Matrix([[2, 1$5]]). Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$2, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);
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Mathematica
CoefficientList[Series[(1 - x + x^5) / (1 - 3 x + 2 x^2 + x^5 - x^6), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)
Formula
G.f.: (1-x+x^5)/(1-3*x+2*x^2+x^5-x^6).