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A365220 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.

%I A365220 #13 Dec 21 2024 18:06:00
%S A365220 1,11,2,22,3,9,4,8,5,7,6,33,99,44,88,55,77,66,101,1001,111,898,121,
%T A365220 888,131,878,141,868,151,858,161,848,171,838,181,828,191,818,404,808,
%U A365220 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
%N A365220 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.
%C A365220 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.
%H A365220 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 A365220 Eric Angelini, <a href="/A365217/a365217.pdf">Go down, go up, go flat integers</a>, Personal blog "Cinquante signes", Aug 2023. [Cached copy]
%e A365220 a(1) + a(2) = 1 + 11 = 12 and 12 is a GUI;
%e A365220 a(2) + a(3) = 11 + 2 = 13 and 13 is a GUI;
%e A365220 a(3) + a(4) = 2 + 22 = 24 and 24 is a GUI;
%e A365220 a(4) + a(5) = 22 + 3 = 25 and 25 is a GUI;
%e A365220 a(5) + a(6) =  3 + 9 = 12 and 12 is a GUI; etc.
%t A365220 a[1]=1;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 A365220 Cf. A365217, A365219.
%K A365220 base,nonn
%O A365220 1,2
%A A365220 _Eric Angelini_, Aug 26 2023
%E A365220 Data corrected by _Giorgos Kalogeropoulos_