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A334293 First quadrisection of Padovan sequence.

%I A334293 #35 May 30 2025 10:33:34
%S A334293 1,0,2,5,16,49,151,465,1432,4410,13581,41824,128801,396655,1221537,
%T A334293 3761840,11584946,35676949,109870576,338356945,1042002567,3208946545,
%U A334293 9882257736,30433357674,93722435101,288627200960,888855064897,2737314167775,8429820731201,25960439030624
%N A334293 First quadrisection of Padovan sequence.
%H A334293 Colin Barker, <a href="/A334293/b334293.txt">Table of n, a(n) for n = 0..1000</a>
%H A334293 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 4.
%H A334293 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,1).
%F A334293 a(n) = A000931(4n).
%F A334293 a(n) = A099529(2n).
%F A334293 a(n) = Sum_{k=0..n} binomial(2*n-k-1, 2*k-1).
%F A334293 a(n) = 2*a(n-1)+3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2 for n>=3.
%F A334293 G.f.: (1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3). - _Colin Barker_, Apr 27 2020
%e A334293 For n=3, a(3) = 2*a(2) + 3*a(1) + a(0) = 2*2 + 3*0 + 1 = 5.
%o A334293 (PARI) Vec((1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3) + O(x^30)) \\ _Colin Barker_, Apr 27 2020
%Y A334293 Quadrisection of A000931.
%Y A334293 Bisection (even part) of A099529.
%Y A334293 Cf. A099098, A091140.
%K A334293 nonn,easy
%O A334293 0,3
%A A334293 _Oboifeng Dira_, Apr 21 2020