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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // G. C. Greubel, Oct 10 2022

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 *)

PARI

Vec(3/(x^2-4*x+1)+O(x^99)) \\ Charles R Greathouse IV, Mar 05 2012

SageMath

[3*chebyshev_U(n-1,2) for n in range(41)] # G. C. Greubel, Oct 10 2022

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

Extensions

Typo in definition corrected by Johannes Boot, Feb 05 2009