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Number of vertices of type B at level n of the hyperbolic Pascal pyramid PP_(4,5).
1

%I #16 Oct 10 2018 10:34:36

%S 0,0,0,1,4,12,33,88,232,609,1596,4180,10945,28656,75024,196417,514228,

%T 1346268,3524577,9227464,24157816,63245985,165580140,433494436,

%U 1134903169,2971215072,7778742048,20365011073,53316291172,139583862444,365435296161,956722026040

%N Number of vertices of type B at level n of the hyperbolic Pascal pyramid PP_(4,5).

%C a(n+2) = A027941(n) = Fibonacci(2n+1) - 1 for n >= 0. - _Georg Fischer_, Oct 09 2018

%H Colin Barker, <a href="/A293064/b293064.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 (2nd line of Table 1).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).

%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 4.

%F From _Colin Barker_, Oct 07 2017: (Start)

%F G.f.: x^3 / ((1 - x)*(1 - 3*x + x^2)).

%F a(n) = -1 + (2^(-n)*((3-sqrt(5))^n*(2+sqrt(5)) + (-2+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5) for n>0.

%F (End)

%t Join[{0}, LinearRecurrence[{4, -4, 1}, {0, 0, 1}, 31]] (* _Jean-François Alcover_, Oct 07 2017 *)

%o (PARI) concat(vector(3), Vec(x^3 / ((1 - x)*(1 - 3*x + x^2)) + O(x^40))) \\ _Colin Barker_, Oct 07 2017

%Y Cf. A293066, A027941, A293070.

%K nonn,easy

%O 0,5

%A _Eric M. Schmidt_, Sep 30 2017