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A077442
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2*n^2 + 7 is a square.
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7
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1, 3, 9, 19, 53, 111, 309, 647, 1801, 3771, 10497, 21979, 61181, 128103, 356589, 746639, 2078353, 4351731, 12113529, 25363747, 70602821, 147830751, 411503397, 861620759, 2398417561, 5021893803, 13979001969, 29269742059, 81475594253
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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).
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REFERENCES
| L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
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.
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.
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FORMULA
| For n>0, a(2n)=A046090(n)+A001653(n)+A001652(n-1); a(2n+1)=A001652(n+1)-A001652(n-1)-A001653(n-1); e.g. 53=21+29+3; 111=119-3-5 - Charlie Marion (charliem(AT)bestweb.net), Aug 14 2003
The same recurrences hold for the odd and even indices respectively : a(n+2)=6*a(n+1)-a(n), a(n+1)=3*a(n)+2*(2*a(n)^2+7)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 11 2007
G.f.: (x+1)^3/(x^2+2*x-1)/(x^2-2*x-1). a(n)= [ -A077985(n)-3*A077985(n-1)+3*A000129(n+1)+A000129(n)]/2. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
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MATHEMATICA
| CoefficientList[Series[(1+3 x+3 x^2+x^3)/ (1-6 x^2+x^4), {x, 0, 50}], x] (* From Harvey P. Dale, Mar 12 2011 *)
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CROSSREFS
| For x an element of A077443 and y the corresponding element of this sequence, the generalized Pell equation x^2 - 2*y^2 = 7 is satisfied.
Cf. A077443.
Sequence in context: A146901 A147477 A146677 * A147455 A146429 A018316
Adjacent sequences: A077439 A077440 A077441 * A077443 A077444 A077445
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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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