A105693 a(n) = Fibonacci(2n+2)-2^n.
0, 1, 4, 13, 39, 112, 313, 859, 2328, 6253, 16687, 44320, 117297, 309619, 815656, 2145541, 5637351, 14799280, 38826025, 101809867, 266865720, 699311581, 1832117599, 4799138368, 12569491809, 32917725667, 86200462408, 225717215989, 591018294423, 1547471885008
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807. See Table 4.
- Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 15, 17, 19.
- Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 22.
- Index entries for linear recurrences with constant coefficients, signature (5,-7,2).
Programs
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Magma
[Fibonacci(2*n+2)-2^n: n in [0..30]]; // Vincenzo Librandi, Apr 21 2011
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Mathematica
Table[Fibonacci[2n+2]-2^n,{n,0,30}] (* or *) LinearRecurrence[{5,-7,2},{0,1,4},40] (* Harvey P. Dale, Jul 21 2016 *) -
PARI
concat(0, Vec(x*(1-x)/((1-2*x)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Sep 12 2016
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PARI
a(n)=fibonacci(2*n+2)-2^n \\ Charles R Greathouse IV, Sep 12 2016
Formula
G.f.: x(1-x)/((1-2x)(1-3x+x^2)).
a(n) = sum{k=0..n+1, binomial(n+1, k+1)*sum{j=0..floor(k/2), F(k-2j)}}.
a(n) = 5*a(n-1)-7*a(n-2)+2*a(n-3) for n>2. - Colin Barker, Sep 12 2016