A293066 Number of vertices at level n of the hyperbolic Pascal pyramid PP_(4,5).

1, 3, 6, 11, 21, 44, 101, 247, 626, 1615, 4201, 10968, 28681, 75051, 196446, 514259, 1346301, 3524612, 9227501, 24157855, 63246026, 165580183, 433494481, 1134903216, 2971215121, 7778742099, 20365011126, 53316291227, 139583862501, 365435296220, 956722026101

Offset

0,2

Links

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

László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (6th line of Table 1).

Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Formula

a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) - a(n-4), n >= 5.

From Colin Barker, Oct 07 2017: (Start)

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

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

(End)

a(n) = 2*n + Fibonacci(2*n - 1). - Ehren Metcalfe, Apr 18 2019

Mathematica

CoefficientList[ Series[(1 - 2x - x^2)/((x - 1)^2 (x^2 - 3x + 1)), {x, 0, 30}], x] (* or *)
LinearRecurrence[{5, -8, 5, -1}, {1, 3, 6, 11}, 31] (* Robert G. Wilson v, Feb 26 2018 *)

Prog

(PARI) Vec((1 - 2*x - x^2) / ((1 - x)^2*(1 - 3*x + x^2)) + O(x^40)) \\ Colin Barker, Oct 07 2017

Crossrefs

Cf. A293070.

Sequence in context: A024495 A360045 A354695 * A104253 A283668 A191581

Adjacent sequences: A293063 A293064 A293065 * A293067 A293068 A293069

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 30 2017

Status

approved