A367635 -id:A367635 - cp's OEIS frontend

A367646 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - GCT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b, while GCT(a,b) is the largest possible generalized Command transform (cf. A367635) where at least one digit of both a and b can be chosen.

0, 1, 2, 14, 35, 78, 21, 103, 92, 53, 28, 60, 146, 132, 71, 44, 30, 73, 66, 102, 41, 17, 6, 82, 150, 129, 117, 26, 98, 29, 111, 20, 8, 16, 97, 166, 95, 164, 113, 72, 109, 88, 186, 105, 166, 115, 54, 109, 68, 164, 83, 131, 100, 89, 81, 179, 62, 158, 137, 56, 131, 70, 87, 79, 156, 65, 131, 80

Offset: 0

Scott R. Shannon, Nov 25 2023

This is a variation of A367578, where one can choose more than one digit from both a(n-2) and a(n-1) to create the largest possible step to a nonnegative number which has not previously appeared to form a(n). If all such numbers have already appeared the smallest possible forward step is chosen, which is just the standard Comma transform of a(n-2) and a(n-1).

It is conjectured that all nonnegative numbers appear in the sequence. After the first 10 million terms the only fixed points are 0, 1, 2, 29, 65, 84, 222, 377, 491, 499, and it is likely no more exist. The first number to appear twice is a(35) = a(44) = 166.

			a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
a(6) = 21 as GCT(a(4),a(5)) = GCT(35,78) = 57, so a(6) = a(5) - 57 = 21, as 21 is nonnegative and not already in the sequence.
a(13) = 132 as GCT(a(11),a(12)) = GCT(60,146) = 14, so a(13) = a(12) - 14 = 132, as 132 is nonnegative and not already in the sequence. This is the first term to differ from A367578.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 10 million terms.

Cf. A367360, A367635, A367578, A005132, A121805.

A367645 Lexicographically earliest sequence of distinct positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms equals the magnitude of the successive differences between the terms.

Original entry on oeis.org

1, 12, 35, 94, 135, 78, 159, 63, 30, 28, 109, 18

Offset: 1

Scott R. Shannon and Eric Angelini, Nov 25 2023

The sequence is finite; after twelve terms 18 is reached which has no following term - see A367341.

			a(6) = 78 as a(5) = 135, and |78 - 135| = 57 which can be formed from the last digit of 135 and the first digit of 78.
a(12) = 18 as a(11) = 109, and |18 - 109| = 91 which can be formed from the last digit of 109 and the first digit of 18. See A367341 for a proof that no following term can exist.

Cf. A121805, A367341, A367635.

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