|
%I M2919 #95 Oct 10 2022 03:55:08
%S 0,3,12,45,168,627,2340,8733,32592,121635,453948,1694157,6322680,
%T 23596563,88063572,328657725,1226567328,4577611587,17083879020,
%U 63757904493,237947738952,888033051315,3314184466308,12368704813917,46160634789360,172273834343523,642934702584732,2399464975995405,8954925201396888,33420235829592147
%N a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.
%C 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
%C Integer values of x that make 9+3*x^2 a perfect square. - _Lorenz H. Menke, Jr._, Mar 26 2008
%C 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
%C a(n) also give the altitude to the middle side of a Super-Heronian Triangle. - _Johannes Boot_, Oct 14 2010
%C 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
%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005320/b005320.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H I. M. Gessel, Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.
%H E. Keith Lloyd, <a href="http://www.jstor.org/stable/3619201">The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles</a>, Math. Gaz. vol 81 (1997), 231-243.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H William H. Richardson, <a href="http://www.math.wichita.edu/~richardson/heronian/heronian.html"> Super-Heronian Triangles</a> from Johannes Boot, Oct 14 2010
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F a(n) = (sqrt(3)/2)*( (2+sqrt(3))^n - (2-sqrt(3))^n ). - _Antonio Alberto Olivares_, Jan 17 2004
%F G.f.: 3*x/(1-4*x+x^2). - _Harvey P. Dale_, Mar 04 2012
%F a(n) = 3*A001353(n). - _R. J. Mathar_, Mar 14 2016
%p A005320:=3*z/(1-4*z+z**2); # _Simon Plouffe_ in his 1992 dissertation
%p 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
%t LinearRecurrence[{4,-1},{0,3},40] (* _Harvey P. Dale_, Mar 04 2012 *)
%o (PARI) Vec(3/(x^2-4*x+1)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 05 2012
%o (Magma) [3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // _G. C. Greubel_, Oct 10 2022
%o (SageMath) [3*chebyshev_U(n-1,2) for n in range(41)] # _G. C. Greubel_, Oct 10 2022
%Y Cf. A001075, A001353, A002194, A003500, A082841.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Typo in definition corrected by _Johannes Boot_, Feb 05 2009
|
