A005320 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3. (Formerly M2919)

0, 3, 12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308, 12368704813917, 46160634789360, 172273834343523, 642934702584732, 2399464975995405, 8954925201396888, 33420235829592147

Offset

0,2

Comments

For n > 1, a(n-1) is the determinant of the n X n band matrix which has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- and subdiagonal. This matrix appears when constructing a natural cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, Feb 14 2007

Integer values of x that make 9+3*x^2 a perfect square. - Lorenz H. Menke, Jr., Mar 26 2008

The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence whose numerators are the terms of this sequence and denominators are A001075. - Clark Kimberling, Aug 27 2008

a(n) also give the altitude to the middle side of a Super-Heronian Triangle. - Johannes Boot, Oct 14 2010

a(n) gives values of y satisfying 3*x^2 - 4*y^2 = 12; corresponding x values are given by A003500. - Sture Sjöstedt, Dec 19 2017

References

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

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

Links

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

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

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

I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5

Tanya Khovanova, Recursive Sequences

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.

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

William H. Richardson, Super-Heronian Triangles from Johannes Boot, Oct 14 2010

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

Formula

a(n) = (sqrt(3)/2)*( (2+sqrt(3))^n - (2-sqrt(3))^n ). - Antonio Alberto Olivares, Jan 17 2004

G.f.: 3*x/(1-4*x+x^2). - Harvey P. Dale, Mar 04 2012

a(n) = 3*A001353(n). - R. J. Mathar, Mar 14 2016

Maple

A005320:=3*z/(1-4*z+z**2); # Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[3, 0]]). Matrix([[4, 1], [ -1, 0]])^n)[1, 2]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 14 2008

Mathematica

LinearRecurrence[{4, -1}, {0, 3}, 40] (* Harvey P. Dale, Mar 04 2012 *)

Prog

(PARI) Vec(3/(x^2-4*x+1)+O(x^99)) \\ Charles R Greathouse IV, Mar 05 2012
(Magma) [3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // G. C. Greubel, Oct 10 2022
(SageMath) [3*chebyshev_U(n-1, 2) for n in range(41)] # G. C. Greubel, Oct 10 2022

Crossrefs

Cf. A001075, A001353, A002194, A003500, A082841.

Sequence in context: A229936 A258626 A064017 * A062561 A128593 A359813

Adjacent sequences: A005317 A005318 A005319 * A005321 A005322 A005323

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Typo in definition corrected by Johannes Boot, Feb 05 2009

Status

approved