A264236 Number of vertices at level n of the hyperbolic Pascal pyramid.

1, 3, 6, 13, 36, 138, 736, 4908, 36351, 280228, 2190651, 17206203, 135357481, 1065387963, 8387050686, 66029196613, 519841755036, 4092692363058, 32221664474776, 253680537891828, 1997222414704551, 15724098193422028, 123795561597659331, 974640390569138163

Offset

0,2

Links

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

László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (6th line of Table 1).

Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Formula

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5).

a(n) = (-3/2 + 9*sqrt(5)/10)*((3 + sqrt(5))/2)^n + (-3/2 - 9*sqrt(5)/10)*((3 - sqrt(5))/2)^n + (7/12 - 3*sqrt(15)/20)*(4 + sqrt(15))^n + (7/12 + 3*sqrt(15)/20)*(4 - sqrt(15))^n + 17/6. (See Németh paper, page 9.)

G.f.: (1 - 9*x + 7*x^2 + 15*x^3 + 3*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 - 8*x + x^2)). [Bruno Berselli, Nov 09 2015]

a(n) = A076765(n-3) + 3*Fibonacci(2*(n-1)) + 3. - Ehren Metcalfe, Apr 18 2019

Mathematica

LinearRecurrence[{12, -37, 37, -12, 1}, {1, 3, 6, 13, 36}, 30] (* Bruno Berselli, Nov 09 2015 *)

Prog

(PARI) Vec((1-9*x+7*x^2+15*x^3+3*x^4)/((1-x)*(1-3*x+x^2)*(1-8*x+x^2)) + O(x^50)) \\ Altug Alkan, Nov 09 2015

Crossrefs

Cf. A076765, A027941, A055588, A264237.

Sequence in context: A062466 A366940 A053564 * A216999 A036781 A370208

Adjacent sequences: A264233 A264234 A264235 * A264237 A264238 A264239

Keyword

nonn,easy

Author

Michel Marcus, Nov 09 2015

Extensions

More terms from Bruno Berselli, Nov 09 2015

Status

approved