A094686 A Fibonacci convolution.

1, 0, 1, 2, 2, 4, 7, 10, 17, 28, 44, 72, 117, 188, 305, 494, 798, 1292, 2091, 3382, 5473, 8856, 14328, 23184, 37513, 60696, 98209, 158906, 257114, 416020, 673135, 1089154, 1762289, 2851444, 4613732, 7465176, 12078909, 19544084, 31622993, 51167078

Offset

0,4

Comments

Convolution of A000045 and A049347.

Diagonal sums of number triangle A116088. - Paul Barry, Feb 04 2006

Let (b(n)) be the p-INVERT of (1,1,0,0,0,0,0,0,...) using p(S) = 1 - S^2; then b(n) = a(n+1) for n >=0. See A292324. - Clark Kimberling, Sep 15 2017

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016 (see 1st column of Table 1 p. 8).

Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785 [cs.IT], 2016 [See Table 2].

Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.

David Broadhurst, Multiple Deligne values: a data mine with empirically tamed denominators, arXiv:1409.7204 [hep-th], 2014 (see p. 10).

Leonard Rozendaal, Pisano word, tesselation, plane-filling fractal, Preprint, 2017.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Formula

G.f.: 1/((1-x-x^2)*(1+x+x^2)).

a(n) = 2*sqrt(3)*Sum_{k=0..n} Fibonacci(k+1)*cos((4*(n-k)+1)*Pi/6)/3.

a(n) = a(n-2) + 2*a(n-3) + a(n-4).

From Paul Barry, Jan 13 2005

a(n) = A005252(n) - (-cos((2*n+1)*Pi/3)/2 - sqrt(3)*sin((2*n+1)*Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/6 + sin((2*n+1)*Pi/6)/2).

a(n) = Sum_{k=0..floor(n/2)} if(mod(n-k, 2)=0, binomial(n-k, k), 0).

a(n) = A093040(n-1) - Fibonacci(n). (End)

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(1+(-1)^(n-k))/2. - Paul Barry, Sep 09 2005

From Paul Barry, Feb 04 2006: (Start)

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

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(3*k,n-k)/C(3*k,k). (End)

2*a(n) = A000045(n+1) + A049347(n). - R. J. Mathar, Feb 13 2020

a(n) = (1/2)*(A000045(n+1) + A049347(n)). - G. C. Greubel, Feb 09 2023

Mathematica

LinearRecurrence[{0, 1, 2, 1}, {1, 0, 1, 2}, 40] (* Jean-François Alcover, Sep 21 2017 *)

Prog

(PARI) Vec(1/((1-x-x^2)*(1+x+x^2)) + O(x^50)) \\ Michel Marcus, Sep 27 2014
(Magma) [(Fibonacci(n+1) +((n+2) mod 3) -1)/2: n in [0..40]]; // G. C. Greubel, Feb 09 2023
(SageMath) [(fibonacci(n+1) + (n+2)%3 - 1)/2 for n in range(41)] # G. C. Greubel, Feb 09 2023

Crossrefs

Cf. A000045, A005252, A049347, A093040, A116088, A292324.

Sequence in context: A082222 A058630 A095092 * A277752 A095054 A316210

Adjacent sequences: A094683 A094684 A094685 * A094687 A094688 A094689

Keyword

easy,nonn

Author

Paul Barry, May 19 2004

Status

approved