A390190 a(n) is the number of square tiles in ring n (n >= 0) symmetric about a vertex of the hyperbolic regular tiling with Schläfli symbol {4,5}.

5, 25, 95, 355, 1325, 4945, 18455, 68875, 257045, 959305, 3580175, 13361395, 49865405, 186100225, 694535495, 2592041755, 9673631525, 36102484345, 134736305855, 502842739075, 1876634650445, 7003695862705, 26138148800375, 97548899338795

Offset

0,1

Comments

For ring 0, the 5 tiles share a vertex. For ring n+1, each tile shares a vertex with a tile of ring n. This tiling is also called the order-5 square tiling of the hyperbolic plane.

Links

Table of n, a(n) for n=0..23.

Wikipedia, Order-5 square tiling.

Index entries for linear recurrences with constant coefficients, signature (4,-1).

Formula

a(n) = c(n)+2*d(n) where:

c(0)=5, and for n>0, c(n) = c(n-1)+d(n-1);

d(0)=0, and for n>0, d(n) = 2*c(n-1)+3*d(n-1).

From Robert A. Russell, Feb 04 2026: (Start)

a(0) = 5, a(1) = 25, a(n) = 4*a(n-1) - a(n-2).

a(n) = 5*A001834(n). (End)

From Stefano Spezia, Feb 05 2026: (Start)

G.f.: 5*(1 + x)/(1 - 4*x + x^2).

E.g.f.: 5*exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). (End)

Mathematica

LinearRecurrence[{4, -1}, {5, 25}, 50] (* Robert A. Russell, Feb 04 2026 *)

Crossrefs

Cf. A119611, A001834.

Sequence in context: A213293 A203184 A073374 * A126878 A203340 A055343

Adjacent sequences: A390187 A390188 A390189 * A390191 A390192 A390193

Keyword

nonn,easy

Author

John Mason, Érika Roldán, and Rosemberg Toalá-Enríquez, Oct 29 2025

Extensions

Definition corrected by John Mason according to suggestion from Robert A. Russell, Feb 04 2026

Status

approved