OFFSET
1,4
COMMENTS
From Artur Jasinski, Mar 07 2010: (Start)
Zeta(2, k/p) + Zeta(2, (p-k)/p) = (Pi/sin((Pi*a(n))/p))*2, where p=2,3,4, k=1..p-1.
This sequence is the odd subset of A003983 for odd p=3,5,7,9,....
Table T(n,k) n, k > 0, T(n,k) = n-k+1, if n >= k, T(n,k) = k-n, if n < k. Table read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). General case A209301. Let m be a natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A004739, for m=2 the result is A004738, for m=3 the result is A209301. - Boris Putievskiy, Jan 24 2013
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
F. Smarandache, Collected Papers, Vol. II
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences
FORMULA
From Boris Putievskiy, Jan 24 2013: (Start)
For the general case,
a(n) = m*v + (2*v-1)*(t*t-n) + t, where t = floor((sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1.
For m=1,
a(n) = v + (2*v-1)*(t*t-n) + t, where t = floor((sqrt(n) - 1/2) + 1 and v = floor((n-1)/t) - t + 1. (End)
EXAMPLE
From Boris Putievskiy, Jan 24 2013: (Start)
The start of the sequence as table:
1, 1, 2, 3, 4, 5, 6, ...
2, 1, 1, 2, 3, 4, 5, ...
3, 2, 1, 1, 2, 3, 4, ...
4, 3, 2, 1, 1, 2, 3, ...
5, 4, 3, 2, 1, 1, 2, ...
6, 5, 4, 3, 2, 1, 1, ...
7, 6, 5, 4, 3, 2, 1, ...
...
The start of the sequence as triangle array read by rows:
1;
1, 1, 2;
2, 1, 1, 2, 3;
3, 2, 1, 1, 2, 3, 4;
4, 3, 2, 1, 1, 2, 3, 4, 5;
5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6;
6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7;
...
Row number r contains 2*r - 1 numbers: r-1, r-2, ..., 1, 1, 2, ..., r. (End)
MATHEMATICA
aa = {}; Do[Do[AppendTo[aa, (p/Pi) ArcSin[Sqrt[1/((1/Pi^2) (Zeta[2, k/p] + Zeta[2, (p - k)/p]))]]], {k, 1, p - 1}], {p, 3, 50, 2}]; Round[N[aa, 50]] (* Artur Jasinski, Mar 07 2010 *)
PROG
(Haskell)
a004739 n = a004739_list !! (n-1)
a004739_list = concat $ map (\n -> [1..n] ++ [n, n-1..1]) [1..]
-- Reinhard Zumkeller, Mar 26 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. Muller
EXTENSIONS
More terms from Patrick De Geest, Jun 15 1998
STATUS
approved