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

1,2

Comments

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

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

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)

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A057027, A057944, A194959.

Sequence in context: A360371 A194970 A194982 * A371247 A361251 A194969

Adjacent sequences: A064575 A064576 A064577 * A064579 A064580 A064581

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 16 2001

Extensions

More terms from Vladeta Jovovic, Oct 18 2001

Status

approved