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A001354 Coordination sequence for hyperbolic tessellation 3^7 (from triangle group (2,3,7)). 2
1, 7, 21, 56, 147, 385, 1008, 2639, 6909, 18088, 47355, 123977, 324576, 849751, 2224677, 5824280, 15248163, 39920209, 104512464, 273617183, 716339085, 1875400072, 4909861131, 12854183321 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M.  On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 CFC Type H.

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 2nd ed., 1993.

C. Series and D. Wright, Non-Euclidean geometry and Indra's pearls, Plus magazine, Jul 12 2011, (see Fig 1a)

A. Stakhov and A. S. Aranson, Hyperbolic Fibonacci and Lucas Functions, Applied Mathematics, 2(1); 2011. - Avi Friedlich, May 28 2015

Wikipedia, Uniform tiling 73

Index entries for linear recurrences with constant coefficients, signature (3,-1).

FORMULA

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

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

G.f.: (1+4*x+x^2)/(1-3*x+x^2). [Colin Barker, Apr 14 2012]

a(n) = A001906(n-1)+4*A001906(n)+A001906(n+1). - R. J. Mathar, Mar 04 2018

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

EXAMPLE

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

MATHEMATICA

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

a[ n_] := Boole[n == 0] + 7 Fibonacci[2 n]; (* Michael Somos, Jun 07 2015 *)

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

LinearRecurrence[{3, -1}, {1, 7, 21}, 30] (* Harvey P. Dale, Oct 24 2020 *)

PROG

(PARI) {a(n) = (n==0) + 7 * fibonacci(2*n)}; /* Michael Somos, Jun 07 2015 */

(MAGMA) [1] cat [Fibonacci(2*n+3)+2*Lucas(2*n-1):n in [1..30]]; // Marius A. Burtea, Jul 31 2019

CROSSREFS

Cf. A001906, A000045.

Sequence in context: A146408 A050894 A146464 * A200930 A088656 A146139

Adjacent sequences:  A001351 A001352 A001353 * A001355 A001356 A001357

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 20 10:11 EDT 2021. Contains 343130 sequences. (Running on oeis4.)