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A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

%I A265278 #27 Jul 20 2024 10:53:59
%S A265278 0,1,2,6,16,40,98,238,576,1392,3362,8118,19600,47320,114242,275806,
%T A265278 665856,1607520,3880898,9369318,22619536,54608392,131836322,318281038,
%U A265278 768398400,1855077840,4478554082,10812186006,26102926096,63018038200,152139002498,367296043198
%N A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).
%H A265278 Colin Barker, <a href="/A265278/b265278.txt">Table of n, a(n) for n = 0..1000</a>
%H A265278 Marika Diepenbroek, Monica Maus, and Alex Stoll, <a href="http://www.valpo.edu/mathematics-statistics/files/2014/09/Pudwell2015.pdf">Pattern Avoidance in Reverse Double Lists</a>, Preprint 2015. See Table 3.
%H A265278 Hannah Golab, <a href="http://danaernst.com/scholarship/GolabThesis.pdf">Pattern avoidance in Cayley permutations</a>, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.
%H A265278 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F A265278 From _Colin Barker_, Apr 12 2016: (Start)
%F A265278 a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n)/2 for n>1.
%F A265278 a(n) = 3*a(n-1)-a(n-2)-a(n-3) for n>4.
%F A265278 (End)
%F A265278 E.g.f.: x + (cosh(sqrt(2)*x) - 1)*exp(x). - _Ilya Gutkovskiy_, Sep 16 2016
%t A265278 Table[2 Fibonacci[n-1, 2] + LucasL[n-1, 2]/2 + KroneckerDelta[n-1] - 1, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)
%t A265278 LinearRecurrence[{3,-1,-1},{0,1,2,6,16},40] (* _Harvey P. Dale_, Mar 18 2018 *)
%o A265278 (PARI) concat(0, Vec(x*(1-x+x^2+x^3)/((1-x)*(1-2*x-x^2)) + O(x^50))) \\ _Colin Barker_, Apr 12 2016
%Y A265278 Agrees with A213667 except for initial terms.
%Y A265278 Cf. A002605, A265106, A265107, A270810.
%K A265278 nonn,easy
%O A265278 0,3
%A A265278 _Alois P. Heinz_, Apr 06 2016