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A078358
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Non-oblong numbers: Complement of A002378.
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15
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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, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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, 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). Because infinite subseries 1/A002378(k+1) is convergent to 1 it means that Sum[1/A078358(n),{n,1,Infinity}] is divergent. [From Artur Jasinski, 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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_Reinhard Zumkeller_, Table of n, a(n) for n = 1..10000
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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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MATHEMATICA
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Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* From Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
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PROG
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(Haskell)
a078358 n = a078358_list !! (n-1)
a078358_list = compl [0..] a002378_list where
compl (u:us) vs'@(v:vs) | u == v = compl us vs
| u /= v = u : compl us vs'
-- Reinhard Zumkeller, May 08 2012
(PARI) a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013
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CROSSREFS
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a(n)=(A077425(n)-1)/4.
Cf. A049068 (subsequence), A144786.
Sequence in context: A039177 A058986 A184431 * A175968 A152012 A173153
Adjacent sequences: A078355 A078356 A078357 * 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, Nov 29 2002
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STATUS
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approved
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