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Search: id:A077444
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| A077444 |
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Numbers n such that (n^2 + 4)/2 is a square. |
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+0
8
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| 2, 14, 82, 478, 2786, 16238, 94642, 551614, 3215042, 18738638, 109216786, 636562078, 3710155682, 21624372014, 126036076402, 734592086398, 4281516441986, 24954506565518, 145445522951122, 847718631141214
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The equation "(n^2 + 4)/2 is a square" is a version of the generalized Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = -4.
Sequence of all positive integers k such that continued fraction [k,k,k,k,k,k,...] belongs to Q(sqrt(2)). - Thomas Baruchel Sep 15 2003
Equivalently, 2*n^2 + 8 is a square.
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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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Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, ; Pell Equation
Eric Weisstein's World of Mathematics, NSW Number
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FORMULA
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a(n) =[ [(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] + [(3+2*Sqrt(2))^(n-1) - 3-2*Sqrt(2))^(n-1)] ] / (2*Sqrt(2))
Recurrence: a(n) = 6*a(n-1) - a(n-2), starting 2, 14.
Offset 0, with a=3+2sqrt(2), b=3-2sqrt(2): a(n)=a^((2n+1)/2)-b^((2n+1)/2). a(n)=2(A001109(n+1)+A001109(n))=(A003499(n+1)-A003499(n))/2=2sqrt(A001108(2n+1)) =sqrt(A003499(2n+1)-2). - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003
Lim. n -> Inf. a(n)/a(n-1) = 5.82842712474619009760 = 3 + 2*Sqrt(2).
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3), a(0) = 2, a(1) = 14; a(3) = 82; a(n) = (1+SQRT(2))^(2N+1) + (1-SQRT(2))^(2N+1)
G.f.: 2*x*(1+x)/(1-6*x+x^2). a(n) = 2*[7*A001109(n)-A001109(n+1)]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
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CROSSREFS
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Equals 2 * A002315.
(A077445(n))^2 - 2*a(n) = 8.
First differences of A001541. Pairwise sums of A001542. Bisection of A002203 and A080039.
Sequence in context: A026291 A102401 A077461 this_sequence A138126 A053141 A036692
Adjacent sequences: A077441 A077442 A077443 this_sequence A077445 A077446 A077447
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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 09 2002
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