A293064 Number of vertices of type B at level n of the hyperbolic Pascal pyramid PP_(4,5).

0, 0, 0, 1, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, 514228, 1346268, 3524577, 9227464, 24157816, 63245985, 165580140, 433494436, 1134903169, 2971215072, 7778742048, 20365011073, 53316291172, 139583862444, 365435296161, 956722026040

Offset

0,5

Comments

a(n+2) = A027941(n) = Fibonacci(2n+1) - 1 for n >= 0. - Georg Fischer, Oct 09 2018

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 (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.

From Colin Barker, Oct 07 2017: (Start)

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

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

(End)

Mathematica

Join[{0}, LinearRecurrence[{4, -4, 1}, {0, 0, 1}, 31]] (* Jean-François Alcover, Oct 07 2017 *)

Prog

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

Crossrefs

Cf. A293066, A027941, A293070.

Sequence in context: A070050 A186025 A027941 * A219092 A135254 A326804

Adjacent sequences: A293061 A293062 A293063 * A293065 A293066 A293067

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 30 2017

Status

approved