Search: id:A011900
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%I A011900
%S A011900 1,3,15,85,493,2871,16731,97513,568345,3312555,19306983,112529341,
%T A011900 655869061,3822685023,22280241075,129858761425,756872327473,
%U A011900 4411375203411,25711378892991,149856898154533,873430010034205
%N A011900 a(n)=6*a(n-1)-a(n-2)-2, with a(0)=1, a(1)=3.
%C A011900 Members of Diophantine pairs.
%C A011900 Solution to b(b-1) = 2a(a-1) in natural numbers; a = a(n), b = b(n) = 
               A046090(n).
%D A011900 Mario Velucchi "The Pell's equation ... an amusing application" in Mathematics 
               and Informatics Quarterly, to appear 1997.
%F A011900 a(n)= (A001653(n)+1)/2.
%F A011900 a(n)=(((1+Sqrt(2))^(2*n-1)-(1-Sqrt(2))^(2*n-1))/Sqrt(8)+1)/2.
%F A011900 a_n = 7[a_(n-1) - a_(n-2)] + a_(n-3); a_(1) = 1, a_(2) = 3, a_(3) = 15. 
               Also a(n) = 1/2 + ( (1-sqrt(2))/(-4*sqrt(2)) )*(3-2*sqrt(2))^n + 
               ( (1+sqrt(2))/(4*sqrt(2)) )*(3+2*sqrt(2))^n. - Antonio Alberto Olivares, 
               Dec 23 2003
%F A011900 Sqrt(2) = Sum_{n=0..inf} 1/a(n); a(n)=a(n-1)+floor(1/(Sqrt(2)-Sum_{k=0..n-1}1/
               a(k))) (n>0) with a(0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jan 25 2004
%F A011900 For n>k, a(n+k)=A001541(n)*A001653(k)-A053141(n-k-1); e.g. 493=99*5-2. 
               For n<=k, a(n+k)=A001541(n)*A001653(k)-A053141(k-n); e.g. 85=3*29-2 
               - Charlie Marion (charliemath(AT)optonline.net), Oct 18 2004
%F A011900 a(n+1)=3*a(n)-1+(8*a(n)^2-8*a(n)+1)^0.5, a(1)=1. - Richard Choulet (richardchoulet(AT)yahoo.fr), 
               Sep 18 2007
%F A011900 G.f.: (1-4*x+x^2)/((1-x)*(1-6*x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 26 2009]
%Y A011900 Cf. A001653, A046090.
%Y A011900 Sequence in context: A005809 A067122 A093593 this_sequence A118342 A084209 
               A127085
%Y A011900 Adjacent sequences: A011897 A011898 A011899 this_sequence A011901 A011902 
               A011903
%K A011900 nonn,easy
%O A011900 0,2
%A A011900 Mario Velucchi (mathchess(AT)velucchi.it)
%E A011900 More terms and comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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