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A023039
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a(n) = 18a(n-1) - a(n-2).
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17
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1, 9, 161, 2889, 51841, 930249, 16692641, 299537289, 5374978561, 96450076809, 1730726404001, 31056625195209, 557288527109761, 10000136862780489, 179445175002939041, 3220013013190122249, 57780789062419261441
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (blekraj(AT)yahoo.com), Jun 25 2002
Chebyshev's polynomials T(n,x) evaluated at x=9.
The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 80*b(n)^2 = +1 with b(n)=A049660(n), n>=0.
Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(5) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 members x of the sequence, 5*x^2 - 5 is a square, A004292(n)^2.
The a(n) are the y-values in the integer solutions of x^2-5y^2=1, see the comment in A060645. - Sture Sjöstedt, Nov 29 2011
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index to sequences with linear recurrences with constant coefficients, signature (18,-1).
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FORMULA
| a(n) ~ 1/2*(sqrt(5) + 2)^(2*n) - Joe Keane (jgk(AT)jgk.org), May 15 2002
Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*Sqrt(5). - Gregory V. Richardson (omomom(AT)hotmail.com), 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[2n*ArcSinh[2]] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008
a(n) = A001077(2*n). - Michael Somos Aug 11 2009
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 01 2010: (Start)
a(n) = 2*A167808(6*n+1) - A167808(6*n+3)
Limit(a(n+k)/a(k), k=infinity) = a(n) + A060645(n)*sqrt(5)
Limit(a(n)/A060645(n), n=infinity) = sqrt(5)
(End)
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MATHEMATICA
| LinearRecurrence[{18, -1}, {1, 9}, 50] (* Sture Sjöstedt, Nov 29 2011 *)
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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
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CROSSREFS
| Bisection of A001077.
Row 2 of array A188645.
Sequence in context: A060348 A062232 A020523 * A159831 A133793 A084874
Adjacent sequences: A023036 A023037 A023038 * A023040 A023041 A023042
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KEYWORD
| nonn,easy,changed
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| More terms from Joe Keane (jgk(AT)jgk.org), May 15 2002
Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
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