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A077443
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Numbers n such that (n^2 - 7)/2 is a square.
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4
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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; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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
| 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
| 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, Pell Equation.
Index to sequences with linear recurrences with constant coefficients, signature (0,6,0,-1)
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FORMULA
| 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.: -x*(x-1)*(3*x^2+8*x+3) / ( (x^2+2*x-1)*(x^2-2*x-1) ) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007
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
a(n) = A000129(n+1)+(-1)^n*A176981(n-1), n>1. - R. J. Mathar, Jul 03 2011
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CROSSREFS
| 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 * A147196 A110225 A065047
Adjacent sequences: A077440 A077441 A077442 * A077444 A077445 A077446
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KEYWORD
| nonn
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AUTHOR
| Gregory V. Richardson (omomom(AT)hotmail.com), Nov 06 2002
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EXTENSIONS
| More terms from Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007
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