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Sum of values of vertices at level n of the hyperbolic Pascal pyramid PP_(4,5).
7

%I #9 Oct 07 2017 10:12:11

%S 1,3,9,29,103,399,1641,6989,30319,132735,583665,2571821,11343223,

%T 50052495,220904217,975041453,4303886431,18997962879,83860441185,

%U 370176644813,1634036256295,7212979975503,31839623961801,140546879747981,620403902366671,2738595239186943

%N Sum of values of vertices at level n of the hyperbolic Pascal pyramid PP_(4,5).

%H Colin Barker, <a href="/A293070/b293070.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 (6th line of Table 2).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-19,14).

%F a(n) = 8*a(n-1) - 19*a(n-2) + 14*a(n-3), n >= 3.

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

%F G.f.: (1 - x)*(1 - 4*x) / ((1 - 2*x)*(1 - 6*x + 7*x^2)).

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

%F (End)

%o (PARI) Vec((1 - x)*(1 - 4*x) / ((1 - 2*x)*(1 - 6*x + 7*x^2)) + O(x^30)) \\ _Colin Barker_, Oct 07 2017

%Y Cf. A293066.

%K nonn,easy

%O 0,2

%A _Eric M. Schmidt_, Oct 03 2017