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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.

%I A365217 #21 Dec 21 2024 18:04:45
%S A365217 10,92,20,82,21,81,31,71,32,70,42,60,43,61,41,62,40,63,50,52,51,53,54,
%T A365217 64,65,72,30,73,74,75,80,76,83,84,85,87,86,90,93,91,94,95,97,96,98,
%U A365217 100,902,110,892,120,882,130,872,140,862,150,852,160,842,170,832,180,822
%N 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.
%C A365217 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.
%H A365217 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2023/08/go-down-go-up-go-flat-integers.html">Go down, go up, go flat integers</a>, Personal blog "Cinquante signes", Aug 2023.
%H A365217 Eric Angelini, <a href="/A365217/a365217.pdf">Go down, go up, go flat integers</a>, Personal blog "Cinquante signes", Aug 2023. [Cached copy]
%e A365217 a(1) + a(2) = 10 + 92 = 102 (a GUI);
%e A365217 a(2) + a(3) = 92 + 20 = 112 (a GUI);
%e A365217 a(3) + a(4) = 20 + 82 = 102 (a GUI);
%e A365217 a(4) + a(5) = 82 + 21 = 103 (a GUI);
%e A365217 a(5) + a(6) = 21 + 81 = 102 (a GUI); etc.
%t A365217 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 *)
%Y A365217 Cf. A336611.
%K A365217 base,nonn
%O A365217 1,1
%A A365217 _Eric Angelini_, Aug 26 2023