A368358 -id:A368358 - cp's OEIS frontend

A368671 For any k >= 0, let P(k) = A368357(k) and P(-k) = A368358(k); for any n > 0, a(n) is the unique k such that P(k) = n.

0, 1, 2, -1, -3, -2, -4, 3, 7, 5, 9, 4, 8, 6, 10, -5, -13, -9, -17, -7, -15, -11, -19, -6, -14, -10, -18, -8, -16, -12, -20, 11, 27, 19, 35, 15, 31, 23, 39, 13, 29, 21, 37, 17, 33, 25, 41, 12, 28, 20, 36, 16, 32, 24, 40, 14, 30, 22, 38, 18, 34, 26, 42, -21

Offset: 1

Rémy Sigrist, Jan 02 2024

This sequence is a bijection from the positive integers to the integers (Z).

			P(2) = A368357(2) = 3, so a(3) = 2.
P(-4) = A368358(4) = 7, so a(7) = -4.

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, PARI program

Cf. A368357, A368358.

PARI
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Conjecture: a(n) = (-1)^(L(n)+1)*(A001045(L(n)+2) - A036044(n)/2 - 1) for n > 0 where L(n) = A000523(n). - Mikhail Kurkov, Dec 13 2024

A368357 Consider the doubly-infinite permutation P defined on page 87 of Davis et al. (1977); sequence gives the terms starting at and to the right of 1.

Original entry on oeis.org

1, 2, 3, 8, 12, 10, 14, 9, 13, 11, 15, 32, 48, 40, 56, 36, 52, 44, 60, 34, 50, 42, 58, 38, 54, 46, 62, 33, 49, 41, 57, 37, 53, 45, 61, 35, 51, 43, 59, 39, 55, 47, 63, 128, 192, 160, 224, 144, 208, 176, 240, 136, 200, 168, 232, 152, 216, 184, 248, 132, 196, 164, 228, 148, 212

Offset: 0

N. J. A. Sloane, Dec 31 2023

nonn

P is a doubly-infinite sequence which is a permutation of the positive integers and contains no increasing or decreasing 4-term arithmetic progression.
A central portion of P, showing terms to the left (see A368358) and right (the present sequence) of the central 1:
..., 18, 28, 20, 24, 16, 7, 5, 6, 4, 1, 2, 3, 8, 12, 10, 14, 9, 13, 11, 15, ...
See the link for a larger portion.

Davis, J. A.; Entringer, R. C.; Graham, R. L.; and Simmons, G. J.; On permutations containing no long arithmetic progressions, Acta Arith. 34 (1977), no. 1, 81-90. The recurrence defining P is given in Fact 6 on page 87.
N. J. A. Sloane, A portion of P showing 511 consecutive terms around 1
N. J. A. Sloane, Maple code

Cf. A003407, A368358 (the left-hand portion, reversed).

cp's OEIS Frontend

A368671 For any k >= 0, let P(k) = A368357(k) and P(-k) = A368358(k); for any n > 0, a(n) is the unique k such that P(k) = n.

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