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A365219 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.

%I A365219 #11 Dec 21 2024 18:26:51
%S A365219 12,18,13,17,14,16,15,25,26,24,19,23,27,34,28,35,29,36,37,38,45,39,46,
%T A365219 47,48,49,152,58,102,68,112,78,122,79,132,69,142,59,162,89,172,108,
%U A365219 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
%N A365219 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.
%C A365219 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.
%H A365219 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 A365219 Eric Angelini, <a href="/A365217/a365217.pdf">Go down, go up, go flat integers</a>, Personal blog "Cinquante signes", Aug 2023. [Cached copy]
%e A365219 a(1) + a(2) = 12 + 18 = 30 and 30 is a GDI;
%e A365219 a(2) + a(3) = 18 + 13 = 31 and 31 is a GDI;
%e A365219 a(3) + a(4) = 13 + 17 = 30 and 30 is a GDI;
%e A365219 a(4) + a(5) = 17 + 14 = 31 and 31 is a GDI;
%e A365219 a(5) + a(6) = 14 + 16 = 30 and 30 is a GDI; etc.
%t A365219 a[1]=12;a[n_]:=a[n]=(k=1;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 A365219 Cf. A365217.
%K A365219 base,nonn
%O A365219 1,1
%A A365219 _Eric Angelini_, Aug 26 2023
%E A365219 Data corrected by _Giorgos Kalogeropoulos_