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Number of vertices at level n of the hyperbolic Pascal pyramid.
7

%I #34 Apr 19 2019 11:45:30

%S 1,3,6,13,36,138,736,4908,36351,280228,2190651,17206203,135357481,

%T 1065387963,8387050686,66029196613,519841755036,4092692363058,

%U 32221664474776,253680537891828,1997222414704551,15724098193422028,123795561597659331,974640390569138163

%N Number of vertices at level n of the hyperbolic Pascal pyramid.

%H Colin Barker, <a href="/A264236/b264236.txt">Table of n, a(n) for n = 0..1000</a>

%H László Németh, <a href="http://arxiv.org/abs/1511.02067">Hyperbolic Pascal pyramid</a>, arXiv:1511.02067 [math.CO], 2015 (6th line of Table 1).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (12,-37,37,-12,1).

%F a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5).

%F a(n) = (-3/2 + 9*sqrt(5)/10)*((3 + sqrt(5))/2)^n + (-3/2 - 9*sqrt(5)/10)*((3 - sqrt(5))/2)^n + (7/12 - 3*sqrt(15)/20)*(4 + sqrt(15))^n + (7/12 + 3*sqrt(15)/20)*(4 - sqrt(15))^n + 17/6. (See Németh paper, page 9.)

%F G.f.: (1 - 9*x + 7*x^2 + 15*x^3 + 3*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 - 8*x + x^2)). [_Bruno Berselli_, Nov 09 2015]

%F a(n) = A076765(n-3) + 3*Fibonacci(2*(n-1)) + 3. - _Ehren Metcalfe_, Apr 18 2019

%t LinearRecurrence[{12, -37, 37, -12, 1}, {1, 3, 6, 13, 36}, 30] (* _Bruno Berselli_, Nov 09 2015 *)

%o (PARI) Vec((1-9*x+7*x^2+15*x^3+3*x^4)/((1-x)*(1-3*x+x^2)*(1-8*x+x^2)) + O(x^50)) \\ _Altug Alkan_, Nov 09 2015

%Y Cf. A076765, A027941, A055588, A264237.

%K nonn,easy

%O 0,2

%A _Michel Marcus_, Nov 09 2015

%E More terms from _Bruno Berselli_, Nov 09 2015