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%I M4005 N1659 #137 Jun 23 2023 16:59:09
%S 1,5,49,485,4801,47525,470449,4656965,46099201,456335045,4517251249,
%T 44716177445,442644523201,4381729054565,43374646022449,
%U 429364731169925,4250272665676801,42073361925598085,416483346590304049
%N a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
%C Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(6). - _Benoit Cloitre_, Feb 14 2004
%C Appears to give all solutions >1 to the equation x^2=ceiling(x*r*floor(x/r)) where r=sqrt(6). - _Benoit Cloitre_, Feb 24 2004
%C a(n) and b(n) (A004189) are the nonnegative proper solutions to the Pell equation a(n)^2 - 6*(2*b(n))^2 = +1, n >= 0. The formula given below by Gregory V. Richardson follows. - _Wolfdieter Lang_, Jun 26 2013
%C a(n) are the integer square roots of (A032528 + 1). They are also the values of m where (A032528(m) - 1) has integer square roots. See A122653 for the integer square roots of (A032528 - 1), and see A122652 for the values of m where (A032528(m) + 1) has integer square roots. - _Richard R. Forberg_, Aug 05 2013
%C a(n) are also the values of m where floor(2m^2/3) has integer square roots, excluding m = 0. The corresponding integer square roots are given by A122652(n). - _Richard R. Forberg_, Nov 21 2013
%C Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 24 = 0. - _Colin Barker_, Feb 09 2014
%C Dickson on page 384 gives the Diophantine equation "24x^2 + 1 = y^2" and later states "y_{n+1} = 10y_n - y_{n-1}" where y_n is this sequence. - _Michael Somos_, Jun 19 2023
%D 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
%D L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 384.
%D L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 374.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.
%H T. D. Noe, <a href="/A001079/b001079.txt">Table of n, a(n) for n=0..200</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and 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 John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H Leonhard Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/29/">De solutione problematum diophanteorum per numeros integros</a>, par. 18.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Robert Phillips, <a href="https://web.archive.org/web/20100713033314/http://www.usca.edu/math/~mathdept/bobp/pdf/polgonal.pdf">Polynomials of the form 1+4ke+4ke^2</a>, 2008.
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F For all members x of the sequence, 6*x^2 -6 is a square. Limit_{n->infinity} a(n)/a(n-1) = 5 + 2*sqrt(6). - _Gregory V. Richardson_, Oct 13 2002
%F a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 10) = A004189(n+1).
%F a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).
%F a(n)*a(n+3) - a(n+1)*a(n+2) = 240. - _Ralf Stephan_, Jun 06 2005
%F Chebyshev's polynomials T(n,x) evaluated at x=5.
%F G.f.: (1-5*x)/(1-10*x+x^2). - _Simon Plouffe_ in his 1992 dissertation
%F a(n)= ((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.
%F a(-n) = a(n).
%F a(n+1) = 5*a(n) + 2*(6*a(n)^2-6)^(1/2) - _Richard Choulet_, Sep 19 2007
%F (sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - _Reinhard Zumkeller_, Mar 12 2008
%F a(n+1) = 2*A054320(n) + 3*A138288(n). - _Reinhard Zumkeller_, Mar 12 2008
%F a(n) = cosh(2*n* arcsinh(sqrt(2))). - _Herbert Kociemba_, Apr 24 2008
%F a(n) = (-1)^n * cos(2*n* arcsin(sqrt(3))). - _Artur Jasinski_, Oct 29 2008
%F a(n) = cos(2*n* arccos(sqrt(3))). - _Artur Jasinski_, Sep 10 2016
%F a(n) = A142238(2n-1) = A041006(2n-1) = A041038(2n-1), for all n > 0. - _M. F. Hasler_, Feb 14 2009
%F 2*a(n)^2 = 3*A122652(n)^2 + 2. - _Charlie Marion_, Feb 01 2013
%F E.g.f.: cosh(2*sqrt(6)*x)*exp(5*x). - _Ilya Gutkovskiy_, Sep 10 2016
%F From _Peter Bala_, Aug 17 2022: (Start)
%F a(n) = (1/2)^n * [x^n] ( 10*x + sqrt(1 + 96*x^2) )^n.
%F The g.f. A(x) satisfies A(2*x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 20*x + 4*x^2) is the g.f. of A098270.
%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p >= 3 and positive integers n and k.
%F Sum_{n >= 1} 1/(a(n) - 3/a(n)) = 1/4.
%F Sum_{n >= 1} (-1)^(n+1)/(a(n) + 2/a(n)) = 1/6.
%F Sum_{n >= 1} 1/(a(n)^2 - 3) = 1/4 - 1/sqrt(24). (End)
%e Pell equation: n = 0: 1^2 - 24*0^2 = +1, n = 1: 5^2 - 6*(1*2)^2 = 1, n = 2: 49^2 - 6*(2*10)^2 = +1. - _Wolfdieter Lang_, Jun 26 2013
%e G.f. = 1 + 5*x + 49*x^2 + 485*x^3 + 4801*x^4 + 47525*x^5 + 470449*x^6 + ...
%p A001079 := proc(n)
%p option remember;
%p if n <= 1 then
%p op(n+1,[1,5]) ;
%p else
%p 10*procname(n-1)-procname(n-2) ;
%p end if;
%p end proc:
%p seq(A001079(n),n=0..20) ; # _R. J. Mathar_, Apr 30 2017
%t Table[(-1)^n Round[N[Cos[2 n ArcSin[Sqrt[3]]], 50]], {n, 0, 20}] (* _Artur Jasinski_, Oct 29 2008 *)
%t a[ n_] := ChebyshevT[n, 5]; (* _Michael Somos_, Aug 24 2014 *)
%t CoefficientList[Series[(1-5*x)/(1-10*x+x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o (PARI) {a(n) = subst(poltchebi(n), 'x, 5)}; /* _Michael Somos_, Sep 05 2006 */
%o (PARI) {a(n) = real((5 + 2*quadgen(24))^n)}; /* _Michael Somos_, Sep 05 2006 */
%o (PARI) {a(n) = n = abs(n); polsym(1 - 10*x + x^2, n)[n+1] / 2}; /* _Michael Somos_, Sep 05 2006 */
%o (Magma) I:=[1,5]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 10 2016
%o (PARI) x='x+O('x^30); Vec((1-5*x)/(1-10*x+x^2)) \\ _G. C. Greubel_, Dec 20 2017
%Y Cf. A004189, A001078, A046173, A046172, A036353, A138281, A004189.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Chebyshev comments from _Wolfdieter Lang_, Nov 08 2002
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