A381003 Lexicographically earliest sequence with a(0) = 0 and a(n) = a(n + a(n)) - a(n - a(n)) > 0.

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

Offset

0,4

Comments

A trivial but not lexicographically earliest solution for this problem is given by a(m) = ceiling(m/2) (A110654).

We assume a(<0) = 0 in this definition but conjecture that a(n) < n for n > 1.

If we iterate recursively k_{n+1} = a(k_{n})+k_{n} starting with k{0}=2, then we will observe a monotone increasing trajectory. An interesting consequence of the existence of this sequence is that if we start a trajectory at some k{m} > 2, such a trajectory would in the majority of cases ( or in all such cases? ) merge into the trajectory which is starting at 2.

Links

Thomas Scheuerle, Table of n, a(n) for n = 0..10000

Thomas Scheuerle, Graph of Sum_{k=0..n} a(k)/n over n, it looks almost like a straight line with slope 0.14.. . Is there any relation to a well known constant?

Thomas Scheuerle, Scatter plot of the first 15000 values, colored by first root ancestor the colormap has period 500.

Thomas Scheuerle, Scatter plot of the first 15000 values, colored by second root ancestor the colormap has period 500.

Thomas Scheuerle, Scatter plot of the first 15000 values, colored by length of the ancestor chain for the first ancestors the colormap has period 15.

Thomas Scheuerle, Scatter plot of the first 15000 values, colored by length of the ancestor chain for the second ancestors the colormap has period 5.

Formula

a(a(n) + n) >= a(n).

Example

a(10) = 5 = a(10 + 5) - a(10 - 5) = 8 - 3.

Crossrefs

Cf. A110654.

Sequence in context: A386500 A393397 A393396 * A333879 A175126 A142884

Adjacent sequences: A381000 A381001 A381002 * A381004 A381005 A381006

Keyword

nonn

Author

Thomas Scheuerle, Feb 11 2025

Status

approved