A357943 - OEIS

%I #34 Oct 23 2022 11:59:34

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C 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.

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

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

%H Scott R. Shannon, <a href="/A357943/b357943.txt">Table of n, a(n) for n = 0..10000</a>

%H Scott R. Shannon, <a href="/A357943/a357943.png">Image of the first 10000000 terms</a>.

%H Scott R. Shannon, <a href="/A357943/a357943_1.png">Frequency distribution of the first 10000000 terms</a>. The maximum x value is 4858, which is the number of times 20 has appeared.

%e 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.

%t 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 *)

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

%K nonn,look

%O 0,3

%A _Scott R. Shannon_, Oct 22 2022