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Search: id:A077443
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| A077443 |
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Numbers n such that (n^2 - 7)/2 is a square. |
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+0
6
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| 3, 5, 13, 27, 75, 157, 437, 915, 2547, 5333, 14845, 31083, 86523, 181165, 504293, 1055907, 2939235, 6154277, 17131117, 35869755, 99847467, 209064253, 581953685, 1218515763, 3391874643, 7102030325, 19769294173, 41393666187
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Lim. n -> Inf. a(n)/a(n-2) = 3 + 2*Sqrt(2) = R1*R2. Lim. k -> Inf. a(2*k-1)/a(2*k) = (9 + 4*Sqrt(2))/7 = R1 (ratio #1). Lim. k -> Inf. a(2*k)/a(2*k-1) = (11 + 6*Sqrt(2))/7 = R2 (ratio #2).
Also gives solutions >3 to the equation x^2-4 = floor(x*r*floor(x/r)) where r=sqrt(2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004
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REFERENCES
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A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
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LINKS
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J. J. O'Connor and E. F. Robertson, History of Pell's Equation
J. P. Robertson, Solving the Generalized Pell Equation
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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The same recurrences hold for the odd and the even indices : a(n+2)=6*a(n+1)-a(n), a(n+1)=3*a(n)+2*(2*a(n)^2-14)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007
O.g.f.: (-1-2*x)/(x^2-2*x-1)+(-2-x)/(x^2+2*x-1) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - Antonio A. Olivares (olivares14031(AT)yahoo.com), Apr 20 2008
If n is even a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^-(1/2)*n) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^-(1/2)*n); if n is odd a(n) = (1/2)*(3+sqrt(2))*(3+2*sqrt(2))^((1/2)n-1/2)) +(1/2)*(3-sqrt(2))*(3-2*sqrt(2))^((1/2)n-1/2)) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Apr 20 2008
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CROSSREFS
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If x = this sequence and y = A077442, the generalized Pell equation x^2 - 2*y^2 = 7 is satisfied.
Cf. A038762, A077442.
Sequence in context: A035082 A005198 A160823 this_sequence A147196 A110225 A065047
Adjacent sequences: A077440 A077441 A077442 this_sequence A077444 A077445 A077446
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KEYWORD
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nonn
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AUTHOR
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Gregory V. Richardson (omomom(AT)hotmail.com), Nov 06 2002
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EXTENSIONS
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More terms from Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007
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