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

0,3

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A264236.

Sequence in context: A290997 A052103 A072168 * A018011 A025208 A245774

Adjacent sequences: A292287 A292288 A292289 * A292291 A292292 A292293

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 13 2017

Status

approved