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A008843
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Squares of NSW numbers (A002315): x^2 such that x^2 - 2.y^2 = -1 for some y.
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3
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1, 49, 1681, 57121, 1940449, 65918161, 2239277041, 76069501249, 2584123765441, 87784138523761, 2982076586042449, 101302819786919521, 3441313796169221281, 116903366249966604049, 3971273138702695316401
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
P. F. Teilhet, Note #2094, L'Intermed. Math., 10 (1903), pp. 235-238.
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LINKS
| Index entries for sequences related to Bernoulli numbers.
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FORMULA
| a(n) = 34a(n-1)-a(n-2)+16 = A002315(n)^2 = 2*A001653(n)^2-1 = 2*A008844(n)-1 = [A046176(n)*sqrt(2) ] = 6*A055792(n+1)-a(n-1)+4 = (6*A055792(n+2)+a(n-1)-20)/35. -Henry Bottomley (se16(AT)btinternet.com), Nov 13 2001
a(n) = sum(k=1, 2*n+1, 2^(k-1)*binomial(4*n+2, 2*k) ). - Zoltan Zachar (zachar(AT)fellner.sulinet.hu), Oct 03 2003
O.g.f.: = -(1+14*x+x^2)/((-1+x)*(1-34*x+x^2)) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007
a(n)=-1/2+(1/2)*sqrt(2)*[17+12*sqrt(2)]^n+(3/4)*[17-12*sqrt(2)]^n-(1/2)*[17-12*sqrt(2)]^n *sqrt(2)+(3/4)*[17+12*sqrt(2)]^n, with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 17 2008
a(n) = -Cosh[(2 n - 1) ArcTanh[Sqrt[2]]])^2 = -1 + (Sinh[(2 n - 1) ArcTanh[Sqrt[2]]])^2 [From Artur Jasinski (grafix(AT)csl.pl), Oct 30 2008]
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MATHEMATICA
| Table[Round[N[ -(Cosh[(2 n - 1) ArcTanh[Sqrt[2]]])^2, 100]], {n, 1, 20}] [From Artur Jasinski (grafix(AT)csl.pl), Oct 30 2008]
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CROSSREFS
| Cf. A002315.
A146313 [From Artur Jasinski (grafix(AT)csl.pl), Oct 30 2008]
Sequence in context: A069327 A088068 A140394 * A145848 A014942 A065785
Adjacent sequences: A008840 A008841 A008842 * A008844 A008845 A008846
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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