A291712 Lexicographically earliest sequence of positive terms such that, for any m and n > 0, if m < n then a(m) != a(n) or a(m+1) != a(n+1), and if n = least k > m such that a(k) = a(m) then m and n have a different parity.

1, 1, 2, 2, 1, 3, 2, 4, 3, 1, 4, 2, 5, 3, 3, 4, 1, 5, 2, 3, 5, 1, 6, 2, 7, 5, 4, 4, 5, 5, 8, 6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, 6, 3, 7, 1, 8, 2, 9, 5, 10, 4, 11, 7, 3, 6, 4, 8, 1, 9, 2, 10, 6, 6, 5, 12, 6, 11, 8, 4, 7, 6, 9, 1, 10, 2, 6, 7, 4, 6, 12, 3, 11, 4

Offset

1,3

Comments

If we drop the constraint "if n = least k > m such that a(k) = a(m) then m and n have a different parity" then we obtain the natural numbers interspersed with 1's: 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, ...

Conjecturally, (a(n), a(n+1)) uniquely runs over all pairs of positive integers (this is the motivation for this sequence).

This sequence has similarities with:

- A226005 whose pairs of consecutive terms run over all pairs of positive integers,

- A290633 whose pairs of consecutive terms (conjecturally) run over all pairs of noncoprime positive integers.

The representation of the first pairs of consecutive terms has nice features.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Colorized scatterplot of the first 10 000 000 pairs of consecutive terms

Rémy Sigrist, PARI program for A291712

Example

a(1) = 1 is suitable.
a(2) = 1 is suitable.
a(3) cannot equal 1 as the pair (1,1) has already been visited.
a(3) = 2 is suitable.
a(4) cannot equal 1 as the previous occurrence of 1 happened at even index.
a(4) = 2 is suitable.
a(5) = 1 is suitable.
a(6) cannot equal 1 as the pair (1,1) has already been visited.
a(6) cannot equal 2 as the previous occurrence of 2 happened at even index.
a(6) = 3 is suitable.

Prog

(PARI) See Links section.

Crossrefs

Cf. A226005, A290633.

Sequence in context: A058741 A339491 A346871 * A074945 A353068 A367292

Adjacent sequences: A291709 A291710 A291711 * A291713 A291714 A291715

Keyword

nonn

Author

Rémy Sigrist, Aug 30 2017

Status

approved