A283834 Number of length-n binary vectors beginning with 0, ending with 1, and avoiding 4 consecutive 0's and 4 consecutive 1's.
1, 0, 1, 2, 4, 6, 12, 22, 41, 74, 137, 252, 464, 852, 1568, 2884, 5305, 9756, 17945, 33006, 60708, 111658, 205372, 377738, 694769, 1277878, 2350385, 4323032, 7951296, 14624712, 26899040, 49475048, 90998801, 167372888, 307846737, 566218426, 1041438052
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 2].
- Index entries for linear recurrences with constant coefficients, signature (0,1,2,3,2,1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Feb 09 2023 -
Mathematica
CoefficientList[Series[1/((1+x)*(1+x^2)*(1-x-x^2-x^3)), {x,0,50}], x] (* Indranil Ghosh, Mar 26 2017 *) -
PARI
Vec(1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) + O(x^50)) \\ Indranil Ghosh, Mar 26 2017
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SageMath
@CachedFunction def b(n): # b = A000073 if (n<3): return (0,0,1)[n] else: return b(n-1) + b(n-2) + b(n-3) def A283834(n): return (1/4)*((-1)^n +i^n*((n+1)%2) -i^(n+3)*(n%2) +2*b(n+2)) [A283834(n) for n in range(41)] # G. C. Greubel, Feb 09 2023
Formula
G.f.: 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)). - Alois P. Heinz, Mar 25 2017
a(n) = (1/4)*((-1)^n + i^n*(n+1 mod 2) - i^(n+3)*(n mod 2) + 2*A000073(n+2)). - G. C. Greubel, Feb 09 2023
Extensions
More terms from Alois P. Heinz, Mar 25 2017