cp's OEIS Frontend

A355072 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number whose sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=1..n-1, whose product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=1..n-1, and whose difference |a(n) - a(n-1)| is distinct from all previous differences, |a(i) - a(i-1)|, i=1..n-1.

%I A355072 #11 Jun 22 2022 07:41:50
%S A355072 0,1,1,3,6,1,5,11,1,9,16,1,10,21,1,13,26,1,17,3,20,1,23,5,28,1,25,46,
%T A355072 1,29,3,32,2,34,3,37,1,40,2,42,1,44,2,46,9,42,7,53,1,49,96,2,55,4,54,
%U A355072 103,1,61,2,59,5,60,116,1,65,2,67,1,69,7,65,126,1,72,5,74,1,73,143,1,77,3,78,155
%N A355072 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number whose sum a(n) + a(n-1) is distinct from all previous sums, a(i) + a(i-1), i=1..n-1, whose product a(n) * a(n-1) is distinct from all previous products, a(i) * a(i-1), i=1..n-1, and whose difference |a(n) - a(n-1)| is distinct from all previous differences, |a(i) - a(i-1)|, i=1..n-1.
%C A355072 For n up to ~35000 the vast majority of terms are concentrated along three lines, the lowest being near the x-axes; see the first linked image. In this same range there are many terms equal to 1; see A355135. Beyond this range the terms no longer fall along the upper-most line and the number of terms equal to 1 greatly diminishes. The reason for this change in behavior is unknown. The remaining upper-most line has a gradient close to 1 and contains multiple fixed points; see A355136 and the second linked image. The sequence it conjectured to contain all the positive integers.
%H A355072 Scott R. Shannon, <a href="/A355072/a355072.png">Image of the first 50000 terms</a>. The green line is y = n.
%H A355072 Scott R. Shannon, <a href="/A355072/a355072_1.png">Image of the first 1000000 terms</a>.
%e A355072 a(3) = 3 as a(2) = 1 and 3+1 = 4, 3*1 = 3, |3-1| = 2, and this product, sum, and difference has not occurred previously.
%e A355072 a(5) = 1 as a(4) = 6 and 1+6 = 7, 1*6 = 6, |1-6| = 5, and this product, sum, and difference has not occurred previously.
%Y A355072 Cf. A355135, A355136, A088177, A008344, A110654.
%K A355072 nonn
%O A355072 0,4
%A A355072 _Scott R. Shannon_, Jun 18 2022