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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

First differences of A082841. Pairwise sums of A001834.

Programs

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 *)

PARI

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

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

Extensions

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