A293065 Number of vertices of type D at level n of the hyperbolic Pascal pyramid PP_(4,5).

0, 0, 0, 1, 3, 7, 16, 38, 94, 239, 617, 1605, 4190, 10956, 28668, 75037, 196431, 514243, 1346284, 3524594, 9227482, 24157835, 63246005, 165580161, 433494458, 1134903192, 2971215096, 7778742073, 20365011099, 53316291199, 139583862472, 365435296190, 956722026070

Offset

0,5

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 (4th 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.

G.f.: x^3*(1 - 2*x) / ((1 - x)^2*(1 - 3*x + x^2)). - Colin Barker, Oct 07 2017

For n > 0, a(n) = Fibonacci(2*n - 5) + n - 3 = Sum_{k=1..n-2} Sum_{j=0..k-1} A001519(j). - Ehren Metcalfe, Apr 18 2019

Mathematica

LinearRecurrence[{5, -8, 5, -1}, {0, 0, 0, 1, 3}, 33] (* a(0)=0 inserted by Georg Fischer, Apr 08 2019 *)

Prog

(PARI) concat(vector(3), Vec(x^3*(1 - 2*x) / ((1 - x)^2*(1 - 3*x + x^2)) + O(x^50))) \\ Colin Barker, Oct 07 2017

Crossrefs

Cf. A293066, A293070.

Sequence in context: A297498 A239040 A323225 * A211278 A364625 A196154

Adjacent sequences: A293062 A293063 A293064 * A293066 A293067 A293068

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 30 2017

Status

approved