A283838 -id:A283838 - cp's OEIS frontend

A283839 Row sums of A283838.

1, 3, 6, 12, 24, 42, 76, 136, 240, 424, 753, 1337, 2388, 4280, 7706, 13940, 25332, 46224, 84696, 155786, 287574, 532624, 989554, 1843744, 3444389, 6450369, 12107004, 22771642, 42913116, 81014528, 153199818, 290152952, 550332614, 1045234672, 1987731140

Offset: 8

N. J. A. Sloane, Mar 25 2017

nonn

a(n) is odd for n >= 8 and n in { A247375 }. - Alois P. Heinz, Mar 26 2017

Alois P. Heinz, Table of n, a(n) for n = 8..1000
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785, 2016

Cf. A247375, A283838.

b:= proc(n, l, c, k) option remember; `if`(n=0, l,
      b(n-1, 1-l, 1, k)+`if`(c=k-1, 0, b(n-1, l, c+1, k)))
    end:
a:= proc(n) option remember; `if`(n<8, 0, a(n-1)+
      add(b(n-2*k-1, 0, 1, k), k=3..floor(n/2)-1))
    end:
seq(a(n), n=8..60);  # Alois P. Heinz, Mar 26 2017

nMax = 60; gf[k_] := gf[k] = x^(2k)(x-x^k)^2 / ((1-x)(1-x^k)(1-2x+x^k)) + O[x]^(nMax+1); a[n_] := Sum[SeriesCoefficient[gf[k], n], {k, 3, Floor[ n/2] - 1}]; Table[a[n], {n, 8, nMax}] (* Jean-François Alcover, Apr 05 2017 *)

More terms from Alois P. Heinz, Mar 26 2017

A283834 Number of length-n binary vectors beginning with 0, ending with 1, and avoiding 4 consecutive 0's and 4 consecutive 1's.

Original entry on oeis.org

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

N. J. A. Sloane, Mar 25 2017

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).

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Feb 09 2023

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

Vec(1/((1+x)*(1+x^2)*(1-x-x^2-x^3)) + O(x^50)) \\ Indranil Ghosh, Mar 26 2017

@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

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

More terms from Alois P. Heinz, Mar 25 2017

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A283839 Row sums of A283838.

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A283834 Number of length-n binary vectors beginning with 0, ending with 1, and avoiding 4 consecutive 0's and 4 consecutive 1's.

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