A292291 Number of vertices of type B at level n of the hyperbolic Pascal pyramid.

0, 0, 0, 3, 12, 36, 99, 264, 696, 1827, 4788, 12540, 32835, 85968, 225072, 589251, 1542684, 4038804, 10573731, 27682392, 72473448, 189737955, 496740420, 1300483308, 3404709507, 8913645216, 23336226144, 61095033219, 159948873516, 418751587332, 1096305888483

Offset

0,4

Links

Colin Barker, Table of n, a(n) for n = 0..1000

László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (2nd 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.

G.f.: 3*x^3 / ((1 - x)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017

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

Mathematica

CoefficientList[Series[3*x^3/((1 - x)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)
LinearRecurrence[{4, -4, 1}, {0, 0, 0, 3}, 40] (* Harvey P. Dale, Oct 25 2017 *)

Prog

(PARI) concat(vector(3), Vec(3*x^3 / ((1 - x)*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Sep 17 2017

Crossrefs

Cf. A264236.

Sequence in context: A334891 A242526 A167667 * A215919 A027327 A290927

Adjacent sequences: A292288 A292289 A292290 * A292292 A292293 A292294

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 13 2017

Status

approved