A004738 Concatenation of sequences (1,2,...,n-1,n,n-1,...,2) for n >= 2.

1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9

Offset: 1

Author

R. Muller

Keywords

• nonn
• easy

Comments

Also concatenation of sequences n,n-1,...,2,1,2,...,n-1,n.

Table T(n,k) n, k > 0, T(n,k) = n-k+1, if n >= k, T(n,k) = k-n+1, 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

Examples

			From _Boris Putievskiy_, Jan 24 2013: (Start)
The start of the sequence as table:
  1, 2, 3, 4, 5, 6, 7, ...
  2, 1, 2, 3, 4, 5, 6, ...
  3, 2, 1, 2, 3, 4, 5, ...
  4, 3, 2, 1, 2, 3, 4, ...
  5, 4, 3, 2, 1, 2, 3, ...
  6, 5, 4, 3, 2, 1, 2, ...
  7, 6, 5, 4, 3, 2, 1, ...
  ...
The start of the sequence as triangle array read by rows:
  1;
  2, 1, 2;
  3, 2, 1, 2, 3;
  4, 3, 2, 1, 2, 3, 4;
  5, 4, 3, 2, 1, 2, 3, 4, 5;
  6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6;
  7, 6, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7;
  ...
Row number r contains 2*r - 1 numbers: r, r-1, ..., 1, 2, ..., r. (End)
		

References

• F. Smarandache, "Numerical Sequences", University of Craiova, 1975; [ See Arizona State University, Special Collection, Tempe, AZ, USA ].

Links

• 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

Crossrefs

Cf. A004737, A004739, A187760, A079813, A209301.

Programs

Maple

A004738 := proc(n)
    local tri ;
    tri := floor(sqrt(n)+1/2) ;
    tri+1-abs(n-1-tri^2) ;
end proc:
seq(A004738(n),n=1..30) ; #R. J. Mathar, Feb 14 2019

Mathematica

row[n_] := Range[n, 1, -1] ~Join~ Range[2, n];
Array[row, 10] // Flatten (* Jean-François Alcover, Apr 19 2020 *)

PARI

a(n)= floor(sqrt(n)+1/2)+1-abs(n-1-(floor(sqrt(n)+1/2)-1/2)^2)

Python

from math import isqrt
def A004738(n): return abs((t:=isqrt(n-1))*(t+1)-n+1)+1 # Chai Wah Wu, Mar 01 2025

Formula

a(n) = floor(sqrt(n) + 1/2) + 1 - abs(n - 1 - (floor(sqrt(n) + 1/2))^2). - Benoit Cloitre, Feb 08 2003

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=2, a(n) = 2*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)

Extensions

More terms from Patrick De Geest, Jun 15 1998