A064578 Inverse permutation to A057027.

1, 2, 3, 4, 6, 5, 7, 9, 10, 8, 11, 13, 15, 14, 12, 16, 18, 20, 21, 19, 17, 22, 24, 26, 28, 27, 25, 23, 29, 31, 33, 35, 36, 34, 32, 30, 37, 39, 41, 43, 45, 44, 42, 40, 38, 46, 48, 50, 52, 54, 55, 53, 51, 49, 47, 56, 58, 60, 62, 64, 66, 65, 63, 61, 59, 57, 67, 69, 71, 73, 75, 77

Offset: 1

Author

N. J. A. Sloane, Oct 16 2001

Keywords

• nonn
• easy

Comments

The sequence is an intra-block permutation of positive integers. - Boris Putievskiy, Mar 13 2024

Examples

			From _Boris Putievskiy_, Mar 13 2024: (Start)
Start of the sequence as a triangular array T(n,k) read by rows:
       k=1   2   3   4   5   6
  n=1:   1;
  n=2:   2,  3;
  n=3:   4,  6,  5;
  n=4:   7,  9, 10,  8;
  n=5:  11, 13, 15, 14, 12;
  n=6:  16, 18, 20, 21, 19, 17;
Row n contains a permutation block of the n numbers (n-1)*n/2+1, (n-1)*n/2+2, ..., (n-1)*n/2+n to themselves.
Subtracting (n-1)*n/2 from each term in row n gives A194959, in which each row is a permutation of 1..n:
  1;
  1, 2;
  1, 3, 2;
  1, 3, 4, 2;
  1, 3, 5, 4, 2;
  1, 3, 5, 6, 4, 2; (End)
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9870
• Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
• Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A057027, A057944, A194959.

Programs

Mathematica

T[n_, k_] := (n - 1)*n/2 + Min[2*k - 1, 2*(n - k + 1)];
Nmax = 6; Table[T[n, k], {n, 1, Nmax}, {k, 1, n}] // Flatten (* Boris Putievskiy, Mar 29 2024 *)

PARI

a(n) = my(A = (sqrtint(8*n) + 1)\2, B = A*(A - 1)/2, C = n - B); B + if(C <= (A+1)\2, 2*C - 1, 2*(A - C + 1)) \\ Mikhail Kurkov, Mar 12 2024

Formula

From Boris Putievskiy, Mar 29 2024: (Start)

a(n) = A057944(n-1) + A194959(n).

T(n,k) = (n-1)*n/2 + min(2*k-1, 2*(n-k+1)), for 1 <= k <= n.

(End)

Extensions

More terms from Vladeta Jovovic, Oct 18 2001