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A145113 Numbers of length n binary words with fewer than 5 0-digits between any pair of consecutive 1-digits. 3

%I #19 May 22 2024 02:10:47

%S 1,2,4,8,16,32,64,127,251,495,975,1919,3775,7424,14598,28702,56430,

%T 110942,218110,428797,842997,1657293,3258157,6405373,12592637,

%U 24756478,48669960,95682628,188107100,369808828,727025020,1429293563,2809917167,5524151707

%N Numbers of length n binary words with fewer than 5 0-digits between any pair of consecutive 1-digits.

%H Vincenzo Librandi, <a href="/A145113/b145113.txt">Table of n, a(n) for n = 0..1000</a>

%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011) # 11.4.2.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,0,0,-1,1).

%F G.f.: (1-x+x^6)/(1-3*x+2*x^2+x^6-x^7).

%e a(7) = 127 = 2^7-1, because 1000001 is the only binary word of length 7 with not less than 5 0-digits between any pair of consecutive 1-digits.

%p a:= n-> (Matrix([[2, 1$6]]). Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0$3, -1, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

%t CoefficientList[Series[(1 - x + x^6) / (1 - 3 x + 2 x^2 + x^6 - x^7), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 06 2013 *)

%Y 5th column of A145111.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Oct 02 2008

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Last modified June 17 18:36 EDT 2024. Contains 373463 sequences. (Running on oeis4.)