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A004739 Concatenation of sequences (1,2,2,...,n-1,n-1,n,n,n-1,n-1,...,2,2,1) for n >= 1. 5
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, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 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 (list; graph; refs; listen; history; text; internal format)
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,....

For the even subset of A003983 see A004737. (End)

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, 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

Cf. A004737, A004738, A004731, A003983, A079813, A187760, A004738, A209301.

Sequence in context: A136605 A165621 A284321 * A156282 A203776 A242357

Adjacent sequences:  A004736 A004737 A004738 * A004740 A004741 A004742

KEYWORD

nonn,easy,changed

AUTHOR

R. Muller

EXTENSIONS

More terms from Patrick De Geest, Jun 15 1998

STATUS

approved

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Last modified October 16 12:52 EDT 2019. Contains 328060 sequences. (Running on oeis4.)