A001352 a(0) = 1, a(1) = 6, a(2) = 24; for n>=3, a(n) = 4a(n-1) - a(n-2). (Formerly M4164 N1731)

1, 6, 24, 90, 336, 1254, 4680, 17466, 65184, 243270, 907896, 3388314, 12645360, 47193126, 176127144, 657315450, 2453134656, 9155223174, 34167758040, 127515808986, 475895477904, 1776066102630, 6628368932616, 24737409627834, 92321269578720, 344547668687046

Offset

0,2

Comments

Also the coordination sequence of a {4,6} tiling of the hyperbolic plane, where there are 6 squares (with vertex angles Pi/3) around every vertex. - toen (tca110(AT)rsphysse.anu.edu.au), May 16 2005

a(n) is related to the almost-equilateral Heronian triangles because it is the area of the Heronian triangle with edge lengths A003500(n)-1, A003500(n)+1 and 4. - Herbert Kociemba, Mar 19 2021

References

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

D. Fortin, B-spline Toeplitz inverse under corner perturbations, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From N. J. A. Sloane, Oct 22 2012

T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-183.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

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

Formula

G.f.: 1+6x/(1-4x+x^2). - R. J. Mathar, Jun 06 2007

a(n) = sqrt(3)*(-(2-sqrt(3))^n+(2+sqrt(3))^n) for n>0. - Colin Barker, Oct 12 2015

Maple

A001352 := proc(n) coeftayl(1+6*x/(1-4*x+x^2), x=0, n) ; end: for n from 0 to 30 do printf("%d, ", A001352(n)) ; od ; # R. J. Mathar, Jun 06 2007
A001352:=(z+1)**2/(1-4*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Join[{1}, LinearRecurrence[{4, -1}, {6, 24}, 30]] (* Harvey P. Dale, Jul 20 2011 *)

Prog

(PARI) Vec((x+1)^2/(x^2-4*x+1) + O(x^40)) \\ Colin Barker, Oct 12 2015

Crossrefs

First differences of A082841. Pairwise sums of A001834.

Sequence in context: A181618 A002919 A006780 * A155602 A179716 A326755

Adjacent sequences: A001349 A001350 A001351 * A001353 A001354 A001355

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Jun 06 2007

Status

approved