Search: id:A078358
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%I A078358
%S A078358 1,3,4,5,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,
%T A078358 31,32,33,34,35,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,53,54,
%U A078358 55,57,58,59,60,61,62,63,64
%N A078358 Complementary numbers to A002378.
%C A078358 The (primitive) period length k(n)=A077427(n) of the (regular) continued 
               fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine 
               equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the 
               approximants of this continued fraction give all solutions. See A077057.
%C A078358 The following sequences all have the same parity: A004737, A006590, A027052, 
               A071028, A071797, A078358, A078446. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), 
               Mar 16 2003
%C A078358 Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) 
               is divergent and can be distributed on two subseries 1/A002378(k+1) 
               and 1/A078358(m). Becuase infinte subseries 1/A002378(k+1) is convergent 
               to 1 that mean that Sum[1/A078358(n),{n,1,Infinity}] is divergent. 
               [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]
%D A078358 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 
               (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H A078358 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%F A078358 4*a(n)+1 is not a square number.
%F A078358 a(n) = ceiling(squareroot(n)) + n -1. - Leroy Quet Jul 06 2007
%Y A078358 a(n)=(A077425(n)-1)/4.
%Y A078358 A144786 [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]
%Y A078358 Sequence in context: A075748 A039177 A058986 this_sequence A152012 A039131 
               A072225
%Y A078358 Adjacent sequences: A078355 A078356 A078357 this_sequence A078359 A078360 
               A078361
%K A078358 nonn,easy
%O A078358 1,2
%A A078358 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 
               2002

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