Search: id:A008843
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%I A008843
%S A008843 1,49,1681,57121,1940449,65918161,2239277041,76069501249,
%T A008843 2584123765441,87784138523761,2982076586042449,101302819786919521,
%U A008843 3441313796169221281,116903366249966604049,3971273138702695316401
%N A008843 Squares of NSW numbers (A002315): x^2 such that x^2 - 2.y^2 = -1 for 
               some y.
%D A008843 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 256.
%D A008843 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli 
               and Euler, Annals Math., 36 (1935), 637-649.
%D A008843 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
%D A008843 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 288.
%D A008843 P. F. Teilhet, Note #2094, L'Intermed. Math., 10 (1903), pp. 235-238.
%H A008843 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%F A008843 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
%F A008843 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
%F A008843 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
%F A008843 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 (ppl(AT)spl.at), Jun 17 2008
%F A008843 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]
%t A008843 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]
%Y A008843 Cf. A002315.
%Y A008843 A146313 [From Artur Jasinski (grafix(AT)csl.pl), Oct 30 2008]
%Y A008843 Sequence in context: A004374 A069327 A088068 this_sequence A145848 A014942 
               A065785
%Y A008843 Adjacent sequences: A008840 A008841 A008842 this_sequence A008844 A008845 
               A008846
%K A008843 nonn
%O A008843 0,2
%A A008843 N. J. A. Sloane (njas(AT)research.att.com).

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