A336611 -id:A336611 - cp's OEIS frontend

A365217 Each term is a "Go down integer" (GDI), but a(n) + a(n+1) is always a "Go up integer" (GUI). More details in the Comments section.

10, 92, 20, 82, 21, 81, 31, 71, 32, 70, 42, 60, 43, 61, 41, 62, 40, 63, 50, 52, 51, 53, 54, 64, 65, 72, 30, 73, 74, 75, 80, 76, 83, 84, 85, 87, 86, 90, 93, 91, 94, 95, 97, 96, 98, 100, 902, 110, 892, 120, 882, 130, 872, 140, 862, 150, 852, 160, 842, 170, 832, 180, 822

Offset: 1

Eric Angelini, Aug 26 2023

The rightmost digit R of a GDI is always smaller than the leftmost digit L of the same GDI. The first such integer is 10, as we need at least two digits for a sound GDI. Accordingly, the R of a GUI is always larger than its L - the first such integer being 12. When R = L we have a "Go flat integer", or GFI. We admit that 0 is the first GFI (followed by 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, etc.) This sequence is the lexicographically earliest of distinct nonnegative terms with this property, starting with a(1) = 10.

			a(1) + a(2) = 10 + 92 = 102 (a GUI);
a(2) + a(3) = 92 + 20 = 112 (a GUI);
a(3) + a(4) = 20 + 82 = 102 (a GUI);
a(4) + a(5) = 82 + 21 = 103 (a GUI);
a(5) + a(6) = 21 + 81 = 102 (a GUI); etc.

Eric Angelini, Go down, go up, go flat integers, Personal blog "Cinquante signes", Aug 2023.
Eric Angelini, Go down, go up, go flat integers, Personal blog "Cinquante signes", Aug 2023. [Cached copy]

Cf. A336611.

a[1]=10;a[n_]:=a[n]=(k=10;While[Last[i=IntegerDigits@k]>=First@i ||MemberQ[Array[a,n-1],k]||First[i1=IntegerDigits[a[n-1]+k]]>=Last@i1,k++];k);Array[a,100] (* Giorgos Kalogeropoulos, Aug 27 2023 *)

A368591 The sequence is both a succession of triples of monotonically increasing numbers and a succession of triples of monotonically increasing digits. This is the lexicographically earliest such sequence starting with a(1) = 0 formed by distinct number-triples.

Original entry on oeis.org

0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 6, 0, 1, 7, 0, 1, 8, 0, 1, 9, 0, 2, 3, 0, 2, 4, 0, 2, 5, 0, 2, 6, 0, 2, 7, 0, 2, 8, 0, 2, 9, 0, 3, 4, 0, 3, 5, 0, 3, 6, 0, 3, 7, 0, 3, 8, 0, 3, 9, 0, 4, 5, 0, 4, 6, 0, 4, 7, 0, 4, 8, 0, 4, 9, 0, 5, 6, 0, 5, 7, 0, 5, 8, 0, 5, 9, 0

Offset: 1

Eric Angelini, Dec 31 2023

When will 2024 appear?
Terms up to n=507:
0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 6, 0, 1, 7, 0, 1, 8, 0, 1, 9,
0, 2, 3, 0, 2, 4, 0, 2, 5, 0, 2, 6, 0, 2, 7, 0, 2, 8, 0, 2, 9, 0, 3, 4,
0, 3, 5, 0, 3, 6, 0, 3, 7, 0, 3, 8, 0, 3, 9, 0, 4, 5, 0, 4, 6, 0, 4, 7,
0, 4, 8, 0, 4, 9, 0, 5, 6, 0, 5, 7, 0, 5, 8, 0, 5, 9, 0, 6, 7, 0, 6, 8,
0, 6, 9, 0, 7, 8, 0, 7, 9, 0, 8, 9, 1, 2, 3, 1, 2, 4, 1, 2, 5, 1, 2, 6,
1, 2, 7, 1, 2, 8, 1, 2, 9, 1, 3, 4, 1, 3, 5, 1, 3, 6, 1, 3, 7, 1, 3, 8,
1, 3, 9, 1, 4, 5, 1, 4, 6, 1, 4, 7, 1, 4, 8, 1, 4, 9, 1, 5, 6, 1, 5, 7,
1, 5, 8, 1, 5, 9, 1, 6, 7, 1, 6, 8, 1, 6, 9, 1, 7, 8, 1, 7, 9, 1, 8, 9,
2, 3, 4, 2, 3, 5, 2, 3, 6, 2, 3, 7, 2, 3, 8, 2, 3, 9, 2, 4, 5, 2, 4, 6,
2, 4, 7, 2, 4, 8, 2, 4, 9, 2, 5, 6, 2, 5, 7, 2, 5, 8, 2, 5, 9, 2, 6, 7,
2, 6, 8, 2, 6, 9, 2, 7, 8, 2, 7, 9, 2, 8, 9, 3, 4, 5, 3, 4, 6, 3, 4, 7,
3, 4, 8, 3, 4, 9, 3, 5, 6, 3, 5, 7, 3, 5, 8, 3, 5, 9, 3, 6, 7, 3, 6, 8,
3, 6, 9, 3, 7, 8, 3, 7, 9, 3, 8, 9, 4, 5, 6, 4, 5, 7, 4, 5, 8, 4, 5, 9,
4, 6, 7, 4, 6, 8, 4, 6, 9, 4, 7, 8, 4, 7, 9, 4, 8, 9, 5, 6, 7, 5, 6, 8,
5, 6, 9, 5, 7, 8, 5, 7, 9, 5, 8, 9, 6, 7, 8, 6, 7, 9, 6, 8, 9, 7, 8, 9,
0, 1, 20,
1, 2, 12, 3, 12, 30,  1, 2, 13, 4, 12, 30,  1, 2, 14, 5, 12, 30,
1, 2, 15, 6, 12, 30,  1, 2, 16, 7, 12, 30,  1, 2, 17, 8, 12, 30,
1, 2, 18, 9, 12, 30,  1, 2, 23, 4, 12, 31,
2, 3, 12, 3, 12, 31,  2, 3, 13, 4, 12, 31,  2, 3, 14, 5, 12, 31,
2, 3, 15, 6, 12, 31,  2, 3, 16, 7, 12, 31,  2, 3, 17, 8, 12, 31,
2, 3, 18, 9, 12, 31,  2, 3, 23, 4, 12, 32,
3, 4, 12, 3, 12, 32,  3, 4, 13, 4, 12, 32,  3, 4, 14, 5, 12, 32,
3, 4, 15, 6, 12, 32,  3, 4, 16, 7, 12, 32,  3, 4, 17, 8, 12, 32,
3, 4, 18, 9, 12, 32,  3, 4, 23, 4, 12, 33

			The first 10 distinct triples of monotonically increasing numbers are: [0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,1,6],[0,1,7],[0,1,8],[0,1,9],[0,2,3],[0,2,4].
The first 10 distinct triples of monotonically increasing numbers using altogether more than 3 digits are: [0,1,20],[1,2,12],[3,12,30],[1,2,13],[4,12,30],[1,2,14],[5,12,30],[1,2,15],[6,12,30],[1,2,16].
The above 10 triples are also a succession of triples of monotonically increasing digits: [0,1,2][0,1,2],[12,3],[12,3][0,1,2],[13,4],[12,3],[0,1,2],[14,5],[12,3][0,1,2],[15,6],[12,3][0,1,2].

Eric Angelini, Table of n, a(n) for n = 1..507
Eric Angelini, Triples for the new year, Personal blog, December 31, 2023.
Eric Angelini, Triples for the new year, local copy of blog post.

Cf. A336611.

More than the usual number of terms are shown (with permission of the editors) in order to reach the interesting terms.

Terms moved to comment to avoid excessive length of DATA by Hugo Pfoertner, Jan 02 2025

cp's OEIS Frontend

A365217 Each term is a "Go down integer" (GDI), but a(n) + a(n+1) is always a "Go up integer" (GUI). More details in the Comments section.

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