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

0,4

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).

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

Mathematica

CoefficientList[Series[1/((1+x)*(1+x^2)*(1-x-x^2-x^3)), {x, 0, 50}], x] (* Indranil Ghosh, Mar 26 2017 *)

Prog

(PARI) Vec(1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) + O(x^50)) \\ Indranil Ghosh, Mar 26 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Feb 09 2023
(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

Crossrefs

Cf. A000073, A094686, A283835, A283836, A283837, A283838.

Sequence in context: A057575 A354580 A196700 * A341582 A370648 A326114

Adjacent sequences: A283831 A283832 A283833 * A283835 A283836 A283837

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 25 2017

Extensions

More terms from Alois P. Heinz, Mar 25 2017

Status

approved