A357943 a(0) = 0; a(1) = 1, a(2) = 2; for n > 2, a(n) is the number of times the term a(n - 1 - a(n-1)) has appeared in the sequence.

0, 1, 2, 1, 1, 3, 1, 1, 5, 5, 5, 1, 3, 3, 3, 6, 3, 5, 5, 5, 5, 1, 7, 1, 1, 9, 5, 9, 8, 8, 9, 9, 1, 4, 2, 10, 4, 10, 4, 2, 2, 3, 3, 4, 4, 4, 7, 4, 7, 7, 7, 7, 7, 7, 8, 8, 7, 9, 9, 9, 9, 9, 9, 9, 4, 11, 4, 11, 9, 12, 12, 12, 12, 12, 12, 12, 12, 9, 13, 2, 13, 2, 6, 8, 8, 8, 13, 8, 6, 3, 3, 8, 9, 9

Offset

0,3

Comments

In the first 100 million terms the longest run of consecutive equal terms is eight : a(69)..a(76) = 12. There is no other run of equal length in this range, and it is unknown if a longer run exists.

Other than the initial three terms, the first time a term exists that is one more than the previous term is a(29) = 8, a(30) = 9. Remarkably the first time two such consecutive terms exists is not until a(60917874) = 5394, a(60917875) = 5395, a(60917876) = 5396. It is unknown if three or more such terms exist.

Note that if the sequence starts with just a(0) = 0, a(1) = 1 then the resulting sequence is A003056.

The sequence is conjectured to contain all positive numbers. See A357944 for the index of where a given number first appears.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Image of the first 10000000 terms.

Scott R. Shannon, Frequency distribution of the first 10000000 terms. The maximum x value is 4858, which is the number of times 20 has appeared.

Example

a(5) = 3 as the term at a(4 - a(4)) = a(4 - 1) = a(3) = 1, and 1 has appeared three times in the sequence.

Mathematica

nn = 83; c[_] = 0; Array[Set[{a[#], c[#]}, {#, 1}] &, 3, 0]; Do[(a[n] = c[#]; c[c[#]]++) &[a[n - a[n - 1] - 1]], {n, 3, nn}]; Array[a, nn, 0] (* Michael De Vlieger, Oct 23 2022 *)

Crossrefs

Cf. A357944, A003056, A357930, A342585, A181391.

Sequence in context: A280698 A217667 A240850 * A194086 A342723 A164659

Adjacent sequences: A357940 A357941 A357942 * A357944 A357945 A357946

Keyword

nonn,look

Author

Scott R. Shannon, Oct 22 2022

Status

approved