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Search: id:A078358
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(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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.
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
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]
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REFERENCES
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O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
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LINKS
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Leroy Quet, Home Page (listed in lieu of email address)
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FORMULA
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4*a(n)+1 is not a square number.
a(n) = ceiling(squareroot(n)) + n -1. - Leroy Quet Jul 06 2007
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CROSSREFS
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a(n)=(A077425(n)-1)/4.
A144786 [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]
Sequence in context: A075748 A039177 A058986 this_sequence A152012 A039131 A072225
Adjacent sequences: A078355 A078356 A078357 this_sequence A078359 A078360 A078361
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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