A292290 Number of vertices of type A at level n of the hyperbolic Pascal pyramid.

0, 0, 3, 6, 12, 27, 66, 168, 435, 1134, 2964, 7755, 20298, 53136, 139107, 364182, 953436, 2496123, 6534930, 17108664, 44791059, 117264510, 307002468, 803742891, 2104226202, 5508935712, 14422580931, 37758807078, 98853840300, 258802713819, 677554301154

Offset: 0

Author

Eric M. Schmidt, Sep 13 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (1st line of Table 1).
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A264236.

Programs

Mathematica

CoefficientList[Series[3*x^2*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)

PARI

concat(vector(2), Vec(3*x^2*(1 - 2*x) / ((1 - x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

Formula

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

From Colin Barker, Sep 17 2017: (Start)

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

a(n) = 3*(1 + (2^(-1-n)*((7-3*sqrt(5))*(3+sqrt(5))^n - (3-sqrt(5))^n*(7+3*sqrt(5)))) / sqrt(5)) for n>0.

(End)

a(n) = 3*(Fibonacci(2*n - 4) + 1) for n > 0. - Ehren Metcalfe, Apr 18 2019