Go down, go up, go flat integers

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

DATA
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, 190, 812, 200, 802, 201, 801, 211, 791, 221, 781, 231, 771, 241, 761, 251, 751, 261, 741, 271, 731, 281, 721, 291, 711, 301, 701, 302, 700, 312, 690, 322, 680, 332, 670, 342, 660, 352,...

COMMENTS
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.

EXAMPLE
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.

XREF
____________________
August 27th update
This idea is now in the OEIS.
And Giorgos Kalogeropoulos was again quick to correct my data and extend the sequence – many thanks, Giorgos!
He sent me the hereunder fractal images of the first 300 and 1000 terms – what a delight!

Our next submissions to the OEIS are obvious variants of the above sequence:

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

DATA
12, 18, 13, 17, 14, 16, 15, 25, 26, 24, 19, 23, 27, 34, 28, 35, 29, 36, 37, 38, 45, 39, 46, 47, 48, 49, 152, 58, 102, 68, 112, 78, 122, 79, 132, 69, 142, 59, 162, 89, 172, 108, 103, 57, 113, 67, 123, 107, 104, 56, 114, 106, 105, 115, 116, 124, 117, 133, 118, 143, 127, 134, 126, 125, 135, 136, 144, 137, 153, 128, 163, 138, 164, 146, 145, 155, 147, 154, 148, 173, 129, 182, ...

COMMENTS
The rightmost digit R of a GUI is always larger than the leftmost digit L of the same GUI. The first such integer is 12, as we need at least two digits for a sound GUI. Accordingly, the R of a GDI is always smaller than its L - the first such integer being 10. 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) = 12.

EXAMPLE
a(1) + a(2) = 12 + 18 = 30 and 30 is a GDI; 
a(2) + a(3) = 18 + 13 = 31 and 31 is a GDI;
a(3) + a(4) = 13 + 17 = 30 and 30 is a GDI;
a(4) + a(5) = 17 + 14 = 31 and 31 is a GDI;
a(5) + a(6) = 14 + 16 = 30 and 30 is a GDI; etc.

XREF
A365217

This sequence was submitted here to the OEIS.


VARIANT #2 NAME
Each term is a "Go flat integer" (GFI), but a(n) + a(n+1) is always a "Go up integer" (GUI). More details in the Comments section.

DATA
1, 11, 2, 22, 3, 9, 4, 8, 5, 7, 6, 33, 99, 44, 88, 55, 77, 66, 101, 1001, 111, 898, 121, 888, 131, 878, 141, 868, 151, 858, 161, 848, 171, 838, 181, 828, 191, 818, 404, 808, 414, 595, 424, 585, 434, 575, 444, 565, 454, 555, 464, 545, 474, 535, 484, 525, 494, 515, 707, 505, 717, 292, 727, 282, 737, 272, 747, 262, 757, 252, 767, 242, 777, 232, 787, 222, 797, 212, 1011, 202, 1021 ...

COMMENTS
The rightmost digit R of a GUI is always larger than the leftmost digit L of the same GUI. The first such integer is 12, as we need at least two digits for a sound GUI. 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) = 1.

EXAMPLE
a(1) + a(2) = 1 + 11 = 12 and 12 is a GUI; 
a(2) + a(3) = 11 + 2 = 13 and 13 is a GUI;
a(3) + a(4) = 2 + 22 = 24 and 24 is a GUI;
a(4) + a(5) = 22 + 3 = 25 and 25 is a GUI;
a(5) + a(6) = 3 + 9 = 12 and 12 is a GUI; etc.

XREF
A365217, A365219.

This sequence was submitted here to the OEIS.


VARIANT #3 NAME
Each term is a "Go flat integer" (GFI), but a(n) + a(n+1) is always a "Go down integer" (GDI). More details in the Comments section.

DATA
1, 9, 11, 99, 101, 909, 2, 8, 22, 88, 3, 7, 33, 77, 4, 6, 44, 66, 5, 55, 505, 515, 191, 111, 292, 121, 181, 131, 171, 141, 161, 151, 252, 262, 242, 272, 232, 282, 222, 383, 323, 393, 212, 494, 313, 595, 333, 373, 343, 363, 353, 454, 464, 444, 474, 434, 484, 424, 606, 404, 616, 414, 626, 4004, 636, 4014, 646, 4024, 656, 4034, ...

COMMENTS
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. 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) = 1.

EXAMPLE
a(1) + a(2) = 1 + 9 = 10 and 10 is a GDI; 
a(2) + a(3) = 9 + 11 = 20 and 20 is a GDI;
a(3) + a(4) = 11 + 99 = 110 and 110 is a GDI;
a(4) + a(5) = 99 + 101 = 200 and 200 is a GDI;
a(5) + a(6) = 101 + 909 = 1010 and 1010 is a GDI; etc.

XREF
A365217, A365219, A365220.

This last variant was submitted here to the OEIS.
Many thanks again to Giorgos Kalogeropoulos for the data corrections (on the four sequences of this page).






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