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A011900
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a(n)=6*a(n-1)-a(n-2)-2, with a(0)=1, a(1)=3.
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12
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1, 3, 15, 85, 493, 2871, 16731, 97513, 568345, 3312555, 19306983, 112529341, 655869061, 3822685023, 22280241075, 129858761425, 756872327473, 4411375203411, 25711378892991, 149856898154533, 873430010034205
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Members of Diophantine pairs.
Solution to b(b-1) = 2a(a-1) in natural numbers; a = a(n), b = b(n) = A046090(n).
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REFERENCES
| Mario Velucchi "The Pell's equation ... an amusing application" in Mathematics and Informatics Quarterly, to appear 1997.
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FORMULA
| a(n)= (A001653(n)+1)/2.
a(n)=(((1+Sqrt(2))^(2*n-1)-(1-Sqrt(2))^(2*n-1))/Sqrt(8)+1)/2.
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
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
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
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
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]
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CROSSREFS
| Cf. A001653, A046090.
Sequence in context: A067122 A202336 A093593 * A118342 A084209 A127085
Adjacent sequences: A011897 A011898 A011899 * A011901 A011902 A011903
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KEYWORD
| nonn,easy
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AUTHOR
| Mario Velucchi (mathchess(AT)velucchi.it)
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EXTENSIONS
| More terms and comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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