login
Coordination sequence for hyperbolic tessellation 3^7 (from triangle group (2,3,7)).
2

%I #79 Sep 08 2022 08:44:29

%S 1,7,21,56,147,385,1008,2639,6909,18088,47355,123977,324576,849751,

%T 2224677,5824280,15248163,39920209,104512464,273617183,716339085,

%U 1875400072,4909861131,12854183321

%N Coordination sequence for hyperbolic tessellation 3^7 (from triangle group (2,3,7)).

%C Conjecture: For k>1, the interlaced polynomials b(2*k-1) = a(k)/7 and b(2*k) = (a(k+1) - a(k)) / 7 are the Fibonacci numbers (A000045). - _Avi Friedlich_, May 25 2015

%H Vincenzo Librandi, <a href="/A001354/b001354.txt">Table of n, a(n) for n = 0..1000</a>

%H Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. <a href="https://doi.org/10.1007/s10801-011-0327-z">On the cyclically fully commutative elements of Coxeter groups</a>, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 CFC Type H.

%H J. H. Conway and N. J. A. Sloane, <a href="http://www.springer.com/us/book/9780387985855">Sphere Packings, Lattices and Groups</a>, Springer, 2nd ed., 1993.

%H C. Series and D. Wright, <a href="http://plus.maths.org/content/non-euclidean-geometry-and-indras-pearls">Non-Euclidean geometry and Indra's pearls</a>, Plus magazine, Jul 12 2011, (see Fig 1a)

%H A. Stakhov and A. S. Aranson, <a href="http://dx.doi.org/10.4236/am.2011.21009">Hyperbolic Fibonacci and Lucas Functions</a>, Applied Mathematics, 2(1); 2011. - _Avi Friedlich_, May 28 2015

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/File:Uniform_tiling_73-t0.png">Uniform tiling 73</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F a(n+1) = 3*a(n) - a(n-1).

%F a(n) = 7*A001906(n), n>0.

%F G.f.: (1 + 4*x + x^2)/(1 - 3*x + x^2). - _Colin Barker_, Apr 14 2012

%F a(n) = A001906(n-1) + 4*A001906(n) + A001906(n+1). - _R. J. Mathar_, Mar 04 2018

%F a(0)=1, and a(n) = F(2*n+3) + 2*L(2*n-1) for n>0, where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - _Rigoberto Florez_, Jul 30 2019

%e G.f. = 1 + 7*x + 21*x^2 + 56*x^3 + 147*x^4 + 385*x^5 + 1008*x^6 + ...

%t CoefficientList[Series[(1+4*x+x^2)/(1-3*x+x^2),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 15 2012 *)

%t a[ n_] := Boole[n == 0] + 7 Fibonacci[2 n]; (* _Michael Somos_, Jun 07 2015 *)

%t Table[If[n == 0, 1, Fibonacci[2*n+3] + 2*LucasL[2*n-1]], {n, 0, 20}] (* _Rigoberto Florez_, Jul 30 2019 *)

%t LinearRecurrence[{3,-1},{1,7,21},30] (* _Harvey P. Dale_, Oct 24 2020 *)

%o (PARI) {a(n) = (n==0) + 7 * fibonacci(2*n)}; /* _Michael Somos_, Jun 07 2015 */

%o (Magma) [1] cat [Fibonacci(2*n+3)+2*Lucas(2*n-1):n in [1..30]]; // _Marius A. Burtea_, Jul 31 2019

%Y Cf. A001906, A000045.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_