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%I #33 Mar 04 2024 15:05:52
%S 0,1,4,13,39,112,313,859,2328,6253,16687,44320,117297,309619,815656,
%T 2145541,5637351,14799280,38826025,101809867,266865720,699311581,
%U 1832117599,4799138368,12569491809,32917725667,86200462408,225717215989,591018294423,1547471885008
%N a(n) = Fibonacci(2n+2)-2^n.
%D Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L. (2018). Enumerations of peaks and valleys on non-decreasing Dyck paths. Discrete Mathematics, 341(10), 2789-2807. See Table 4.
%H Colin Barker, <a href="/A105693/b105693.txt">Table of n, a(n) for n = 0..1000</a>
%H Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 22.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,2).
%F 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)}}.
%F a(n) = A258109(n+1) + A001906(n), n>1. - _Yuriy Sibirmovsky_, Sep 12 2016
%F a(n) = 5*a(n-1)-7*a(n-2)+2*a(n-3) for n>2. - _Colin Barker_, Sep 12 2016
%t Table[Fibonacci[2n+2]-2^n,{n,0,30}] (* or *) LinearRecurrence[{5,-7,2},{0,1,4},40] (* _Harvey P. Dale_, Jul 21 2016 *)
%o (Magma) [Fibonacci(2*n+2)-2^n: n in [0..30]]; // _Vincenzo Librandi_, Apr 21 2011
%o (PARI) concat(0, Vec(x*(1-x)/((1-2*x)*(1-3*x+x^2)) + O(x^40))) \\ _Colin Barker_, Sep 12 2016
%o (PARI) a(n)=fibonacci(2*n+2)-2^n \\ _Charles R Greathouse IV_, Sep 12 2016
%Y Cf. A000045, A061667, A001906, A258109.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 17 2005
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