Search: id:A023039
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%I A023039
%S A023039 1,9,161,2889,51841,930249,16692641,299537289,5374978561,96450076809,
%T A023039 1730726404001,31056625195209,557288527109761,10000136862780489,
%U A023039 179445175002939041,3220013013190122249,57780789062419261441
%N A023039 a(n) = 18a(n-1) - a(n-2).
%C A023039 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
%C A023039 Chebyshev's polynomials T(n,x) evaluated at x=9.
%C A023039 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.
%C A023039 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
%C A023039 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
%H A023039 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A023039 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A023039 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A023039 a(n) ~ 1/2*(sqrt(5) + 2)^(2*n) - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F A023039 For all members x of the sequence, 5*x^2 - 5 is a square. Lim. n-> Inf. 
               a(n)/a(n-1) = phi^6 = 9 + 4*Sqrt(5). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A023039 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).
%F A023039 a(n) = sqrt(80*A049660(n)^2 + 1) (cf. Richardson comment).
%F A023039 a(n) = ((9+4*sqrt(5))^n + (9-4*sqrt(5))^n)/2.
%F A023039 G.f.: (1-9*x)/(1-18*x+x^2).
%F A023039 a(n) = Cosh[2n*ArcSinh[2]] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%e A023039 1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x^4 + 930249*x^5 + 16692641*x^6 
               + ... - Michael Somos Aug 11 2009
%o A023039 (PARI) {a(n) = fibonacci(6*n) / 2 + fibonacci(6*n - 1)} - Michael Somos 
               Aug 11 2009
%Y A023039 Bisection of A001077.
%Y A023039 A001077(2*n) = a(n). - Michael Somos Aug 11 2009
%Y A023039 Sequence in context: A060348 A062232 A020523 this_sequence A159831 A133793 
               A084874
%Y A023039 Adjacent sequences: A023036 A023037 A023038 this_sequence A023040 A023041 
               A023042
%K A023039 nonn
%O A023039 0,2
%A A023039 David W. Wilson (davidwwilson(AT)comcast.net)
%E A023039 More terms from Joe Keane (jgk(AT)jgk.org), May 15 2002
%E A023039 Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 08 2002

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