This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105693 #42 Mar 05 2025 02:03:27
%S A105693 0,1,4,13,39,112,313,859,2328,6253,16687,44320,117297,309619,815656,
%T A105693 2145541,5637351,14799280,38826025,101809867,266865720,699311581,
%U A105693 1832117599,4799138368,12569491809,32917725667,86200462408,225717215989,591018294423,1547471885008
%N A105693 a(n) = Fibonacci(2n+2)-2^n.
%H A105693 Colin Barker, <a href="/A105693/b105693.txt">Table of n, a(n) for n = 0..1000</a>
%H A105693 E. Czabarka et al, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Disc. Math. 341 (2018) 2789-2807. See Table 4.
%H A105693 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 15, 17, 19.
%H A105693 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 A105693 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,2).
%F A105693 G.f.: x(1-x)/((1-2x)(1-3x+x^2)).
%F A105693 a(n) = sum{k=0..n+1, binomial(n+1, k+1)*sum{j=0..floor(k/2), F(k-2j)}}.
%F A105693 a(n) = A258109(n+1) + A001906(n), n>1. - _Yuriy Sibirmovsky_, Sep 12 2016
%F A105693 a(n) = 5*a(n-1)-7*a(n-2)+2*a(n-3) for n>2. - _Colin Barker_, Sep 12 2016
%t A105693 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 A105693 (Magma) [Fibonacci(2*n+2)-2^n: n in [0..30]]; // _Vincenzo Librandi_, Apr 21 2011
%o A105693 (PARI) concat(0, Vec(x*(1-x)/((1-2*x)*(1-3*x+x^2)) + O(x^40))) \\ _Colin Barker_, Sep 12 2016
%o A105693 (PARI) a(n)=fibonacci(2*n+2)-2^n \\ _Charles R Greathouse IV_, Sep 12 2016
%Y A105693 Cf. A000045, A061667, A001906, A258109.
%K A105693 easy,nonn
%O A105693 0,3
%A A105693 _Paul Barry_, Apr 17 2005