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.

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, 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, 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

Offset: 0

Author

Scott R. Shannon, Jun 18 2022

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.
• Scott R. Shannon, Image of the first 1000000 terms.

Crossrefs

Cf. A355135, A355136, A088177, A008344, A110654.