A124280 -id:A124280 - cp's OEIS frontend

A130137 Number of Fibonacci binary words of length n having no 0110 subword. A Fibonacci binary word is a binary word having no 00 subword.

1, 2, 3, 5, 7, 11, 16, 25, 37, 57, 85, 130, 195, 297, 447, 679, 1024, 1553, 2345, 3553, 5369, 8130, 12291, 18605, 28135, 42579, 64400, 97449, 147405, 223033, 337389, 510466, 772227, 1168337, 1767487, 2674063, 4045440, 6120353, 9259217, 14008193

Offset: 0

Emeric Deutsch, May 13 2007

			a(4)=7 because from the 8 Fibonacci binary words of length 4 only 0110 does not qualify.

Michael De Vlieger, Table of n, a(n) for n = 0..5560
Michael A. Allen, Combinations without specified separations and restricted-overlap tiling with combs, arXiv:2210.08167 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1).

Cf. A130136.

a[0]:=1: a[1]:=2: a[2]:=3: a[3]:=5: for n from 4 to 45 do a[n]:=a[n-1]+a[n-2]-a[n-3]+a[n-4] od: seq(a[n],n=0..45);

LinearRecurrence[{1, 1, -1, 1}, {1, 2, 3, 5}, 40] (* Jean-François Alcover, Aug 25 2021 *)

G.f.: (1+z+z^3)/(1-z-z^2+z^3-z^4).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4); a(0)=1, a(1)=2, a(2)=3, a(3)=5.
a(n) = A130136(n,0).
a(n) = A124280(n)+A124280(n-1)+A124280(n-3). - R. J. Mathar, Mar 14 2025

A124279 Riordan array (1/(1-x),x(1-x+x^2)/(1-x)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 8, 7, 5, 1, 1, 1, 6, 12, 13, 9, 6, 1, 1, 1, 7, 17, 22, 19, 11, 7, 1, 1, 1, 8, 23, 35, 35, 26, 13, 8, 1, 1, 1, 9, 30, 53

Offset: 0

Paul Barry, Oct 24 2006

Row sums are A005314. Diagonal sums are A124280. T(2n,n) is A002426.

Reversal of A124445. - Paul Barry, Nov 01 2006

			Triangle begins
1,
1, 1,
1, 1, 1,
1, 2, 1, 1,
1, 3, 3, 1, 1,
1, 4, 5, 4, 1, 1,
1, 5, 8, 7, 5, 1, 1

Number triangle T(n,k)=sum{j=0..n-k, C(j,n-k-j)C(k,n-k-j)}

A124281 Expansion of 1/(1-x-2x^2+2x^3-2*x^4).

Original entry on oeis.org

1, 1, 3, 3, 9, 11, 29, 39, 93, 135, 301, 463, 981, 1575, 3213, 5327, 10565, 17943, 34845, 60255, 115189, 201895, 381453, 675375, 1264869, 2256503, 4198397, 7532415, 13945941, 25126983, 46350829, 83777743, 154117317, 279225111

Offset: 0

Paul Barry, Oct 24 2006

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,2).

Cf. A124280.

a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(j,n-2k-j)C(k,n-2k-j)*2^k}}

cp's OEIS Frontend

A130137 Number of Fibonacci binary words of length n having no 0110 subword. A Fibonacci binary word is a binary word having no 00 subword.

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A124279 Riordan array (1/(1-x),x(1-x+x^2)/(1-x)).

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A124281 Expansion of 1/(1-x-2x^2+2x^3-2*x^4).

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cp's OEIS Frontend

A130137 Number of Fibonacci binary words of length n having no 0110 subword. A Fibonacci binary word is a binary word having no 00 subword.

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Author

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Maple

Mathematica

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A124279 Riordan array (1/(1-x),x(1-x+x^2)/(1-x)).

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Author

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Formula

A124281 Expansion of 1/(1-x-2*x^2+2*x^3-2*x^4).

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A124281 Expansion of 1/(1-x-2x^2+2x^3-2*x^4).