|
%I #37 Nov 29 2023 07:00:10
%S 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,
%T 5,6,6,5,4,3,2,1,1,2,3,4,5,6,7,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,8,7,6,5,
%U 4,3,2,1,1,2,3,4,5,6,7,8,9,9,8,7,6,5,4,3,2,1,1,2,3,4,5,6,7,8,9,10,10,9,8,7
%N Concatenation of sequences (1,2,2,...,n-1,n-1,n,n,n-1,n-1,...,2,2,1) for n >= 1.
%C From _Artur Jasinski_, Mar 07 2010: (Start)
%C 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.
%C This sequence is the odd subset of A003983 for odd p=3,5,7,9,....
%C For the even subset of A003983 see A004737. (End)
%C 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
%H Reinhard Zumkeller, <a href="/A004739/b004739.txt">Table of n, a(n) for n = 1..10000</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [Of] Integer Sequences And Pairing Functions</a>, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/CP2.pdf">Collected Papers, Vol. II</a>
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheSequences.html">Smarandache Sequences</a>
%F From _Boris Putievskiy_, Jan 24 2013: (Start)
%F For the general case,
%F 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.
%F For m=1,
%F 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)
%e From _Boris Putievskiy_, Jan 24 2013: (Start)
%e The start of the sequence as table:
%e 1, 1, 2, 3, 4, 5, 6, ...
%e 2, 1, 1, 2, 3, 4, 5, ...
%e 3, 2, 1, 1, 2, 3, 4, ...
%e 4, 3, 2, 1, 1, 2, 3, ...
%e 5, 4, 3, 2, 1, 1, 2, ...
%e 6, 5, 4, 3, 2, 1, 1, ...
%e 7, 6, 5, 4, 3, 2, 1, ...
%e ...
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 1, 1, 2;
%e 2, 1, 1, 2, 3;
%e 3, 2, 1, 1, 2, 3, 4;
%e 4, 3, 2, 1, 1, 2, 3, 4, 5;
%e 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6;
%e 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7;
%e ...
%e Row number r contains 2*r - 1 numbers: r-1, r-2, ..., 1, 1, 2, ..., r. (End)
%t 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 *)
%o (Haskell)
%o a004739 n = a004739_list !! (n-1)
%o a004739_list = concat $ map (\n -> [1..n] ++ [n,n-1..1]) [1..]
%o -- _Reinhard Zumkeller_, Mar 26 2011
%Y Cf. A004737, A004738, A004731, A003983, A079813, A187760, A004738, A209301.
%K nonn,easy
%O 1,4
%A R. Muller
%E More terms from _Patrick De Geest_, Jun 15 1998
|
