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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.
3
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
0,4
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.
LINKS
Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jun 18 2022
STATUS
approved