%I #24 May 22 2019 23:42:30
%S 0,0,0,1,3,7,16,38,94,239,617,1605,4190,10956,28668,75037,196431,
%T 514243,1346284,3524594,9227482,24157835,63246005,165580161,433494458,
%U 1134903192,2971215096,7778742073,20365011099,53316291199,139583862472,365435296190,956722026070
%N Number of vertices of type D at level n of the hyperbolic Pascal pyramid PP_(4,5).
%H Colin Barker, <a href="/A293065/b293065.txt">Table of n, a(n) for n = 0..1000</a>
%H László Németh, <a href="https://arxiv.org/abs/1701.06022">Pascal pyramid in the space H^2 x R</a>, arXiv:1701.06022 [math.CO], 2017 (4th line of Table 1).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,5,-1).
%F a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) - a(n-4), n >= 5.
%F G.f.: x^3*(1 - 2*x) / ((1 - x)^2*(1 - 3*x + x^2)). - _Colin Barker_, Oct 07 2017
%F For n > 0, a(n) = Fibonacci(2*n - 5) + n - 3 = Sum_{k=1..n-2} Sum_{j=0..k-1} A001519(j). - _Ehren Metcalfe_, Apr 18 2019
%t LinearRecurrence[{5, -8, 5, -1}, {0, 0, 0, 1, 3}, 33] (* a(0)=0 inserted by _Georg Fischer_, Apr 08 2019 *)
%o (PARI) concat(vector(3), Vec(x^3*(1 - 2*x) / ((1 - x)^2*(1 - 3*x + x^2)) + O(x^50))) \\ _Colin Barker_, Oct 07 2017
%Y Cf. A293066, A293070.
%K nonn,easy
%O 0,5
%A _Eric M. Schmidt_, Sep 30 2017
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