|
| |
|
|
A078358
|
|
Complementary numbers to A002378.
|
|
12
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| 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]
|
|
|
REFERENCES
| O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
|
|
|
FORMULA
| 4*a(n)+1 is not a square number.
a(n) = ceiling(squareroot(n)) + n -1. - Leroy Quet Jul 06 2007
|
|
|
MATHEMATICA
| 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 *)
|
|
|
CROSSREFS
| a(n)=(A077425(n)-1)/4.
A144786 [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]
Sequence in context: A039177 A058986 A184431 * A175968 A152012 A173153
Adjacent sequences: A078355 A078356 A078357 * A078359 A078360 A078361
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
|
| |
|
|
