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 A023039 a(n) = 18*a(n-1) - a(n-2). 29
 1, 9, 161, 2889, 51841, 930249, 16692641, 299537289, 5374978561, 96450076809, 1730726404001, 31056625195209, 557288527109761, 10000136862780489, 179445175002939041, 3220013013190122249, 57780789062419261441 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The primitive Heronian triangle 3*a(n) +- 2, 4*a(n) has the latter side cut into 2*a(n) +- 3 by the corresponding altitude and has area 10*a(n)*A060645(n). - Lekraj Beedassy, Jun 25 2002 Chebyshev polynomials T(n,x) evaluated at x=9. {a(n)} gives all (unsigned, integer) solutions of Pell equation a(n)^2 - 80*b(n)^2 = +1 with b(n) = A049660(n), n >= 0. {a(n)} gives all possible solutions for x in Pell equation x^2 - D*y^2 = 1 for D=5, D=20 and D=80. The corresponding values for y are A060645 (D=5), A207832 (D=20) and A049660 (D=80). - Herbert Kociemba, Jun 05 2022 Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 14 2004 Appears to give all solutions > 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 24 2004 For all terms x of the sequence, 5*x^2 - 5 is a square, A004292(n)^2. The a(n) are the x-values in the nonnegative integer solutions of x^2 - 5y^2 = 1, see A060645(n) for the corresponding y-values. - Sture Sjöstedt, Nov 29 2011 Rightmost digits alternate repeatedly: 1 and 9 in fact, a(2) = 18*9 - 1 == 1 (mod 10); a(3) = 18*1 - 9 == 9 (mod 10) therefore a(2n) == 1 (mod 10), a(2n+1) == 9 (mod 10). - Carmine Suriano, Oct 03 2013 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..750 (terms 0..200 from Vincenzo Librandi) Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (18,-1). FORMULA a(n) ~ (1/2)*(sqrt(5) + 2)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002 Limit_{n->infinity} a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5). - Gregory V. Richardson, Oct 13 2002 a(n) = T(n, 9) = (S(n, 18) - S(n-2, 18))/2, with S(n, x) := U(n, x/2) and T(n, x), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 18)=A049660(n+1). a(n) = sqrt(80*A049660(n)^2 + 1) (cf. Richardson comment). a(n) = ((9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/2. G.f.: (1 - 9*x)/(1 - 18*x + x^2). a(n) = cosh(2*n*arcsinh(2)). - Herbert Kociemba, Apr 24 2008 a(n) = A001077(2*n). - Michael Somos, Aug 11 2009 From Johannes W. Meijer, Jul 01 2010: (Start) a(n) = 2*A167808(6*n+1) - A167808(6*n+3). Limit_{k->infinity} a(n+k)/a(k) = a(n) + A060645(n)*sqrt(5). Limit_{n->infinity} a(n)/A060645(n) = sqrt(5). (End) a(n) = (1/2)*A087215(n) = (1/2)*(sqrt(5) + 2)^(2*n) + (1/2)*(sqrt(5) - 2)^(2*n). Sum_{n >= 1} 1/( a(n) - 5/a(n) ) = 1/8. Compare with A005248, A002878 and A075796. - Peter Bala, Nov 29 2013 a(n) = 2*A115032(n-1) - 1 = S(n, 18) - 9*S(n-1, 18), with A115032(-1) = 1, and see the above formula with S(n, 18) using its recurrence. - Wolfdieter Lang, Aug 22 2014 a(n) = A128052(3n). - A.H.M. Smeets, Oct 02 2017 a(n) = A049660(n+1) - 9*A049660(n). - R. J. Mathar, May 24 2018 a(n) = hypergeom([n, -n], [1/2], -4). - Peter Luschny, Jul 26 2020 a(n) = L(6*n)/2 for L(n) the Lucas sequence A000032(n). - Greg Dresden, Dec 07 2021 a(n) = cosh(6*n*arccsch(2)). - Peter Luschny, May 25 2022 EXAMPLE G.f. = 1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x4 + 930249*x^5 + 16692641*x^6 + ... MAPLE a := n -> hypergeom([n, -n], [1/2], -4): seq(simplify(a(n)), n=0..16); # Peter Luschny, Jul 26 2020 MATHEMATICA LinearRecurrence[{18, -1}, {1, 9}, 50] (* Sture Sjöstedt, Nov 29 2011 *) CoefficientList[Series[(1-9*x)/(1-18*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *) PROG (PARI) {a(n) = fibonacci(6*n) / 2 + fibonacci(6*n - 1)}; /* Michael Somos, Aug 11 2009 */ (Magma) I:=[1, 9]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 13 2012 (PARI) x='x+O('x^30); Vec((1-9*x)/(1-18*x+x^2)) \\ G. C. Greubel, Dec 19 2017 CROSSREFS Cf. A001077, A115032. Row 2 of array A188645. Row 4 of A322790. Sequence in context: A062232 A337627 A020523 * A243682 A337152 A159831 Adjacent sequences: A023036 A023037 A023038 * A023040 A023041 A023042 KEYWORD nonn,easy AUTHOR EXTENSIONS Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002 Sture Sjöstedt's comment corrected and reformulated by Wolfdieter Lang, Aug 24 2014 STATUS approved

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Last modified March 28 05:31 EDT 2023. Contains 361577 sequences. (Running on oeis4.)