A390190 - OEIS

%I #31 Feb 05 2026 16:20:42

%S 5,25,95,355,1325,4945,18455,68875,257045,959305,3580175,13361395,

%T 49865405,186100225,694535495,2592041755,9673631525,36102484345,

%U 134736305855,502842739075,1876634650445,7003695862705,26138148800375,97548899338795

%N 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}.

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Order-5_square_tiling">Order-5 square tiling</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).

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

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

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

%F From _Robert A. Russell_, Feb 04 2026: (Start)

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

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

%F From _Stefano Spezia_, Feb 05 2026: (Start)

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

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

%t LinearRecurrence[{4, -1}, {5, 25}, 50] (* _Robert A. Russell_, Feb 04 2026 *)

%Y Cf. A119611, A001834.

%K nonn,easy

%O 0,1

%A _John Mason_, Érika Roldán, and Rosemberg Toalá-Enríquez, Oct 29 2025

%E Definition corrected by _John Mason_ according to suggestion from _Robert A. Russell_, Feb 04 2026