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A370408 Lexicographically earliest sequence of positive integers such that no three equal terms appear at distinct indices that are the side lengths of a triangle.

%I A370408 #41 Mar 06 2024 03:48:31
%S A370408 1,1,1,2,1,2,3,1,3,2,4,4,1,5,5,2,3,6,6,7,1,7,4,8,8,2,3,9,5,9,10,10,11,
%T A370408 1,4,11,6,12,12,13,13,2,7,3,5,14,14,15,8,15,16,16,17,17,1,6,18,4,9,18,
%U A370408 19,19,10,20,7,20,21,2,11,21,3,22,22,5,8,23,12,23,24,24,13,25,25,26,26,27,27,28,1,9,28,29,4
%N A370408 Lexicographically earliest sequence of positive integers such that no three equal terms appear at distinct indices that are the side lengths of a triangle.
%C A370408 In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z.
%C A370408 So if x < y and a(x) = a(y) = t then we cannot have a(z) = t for any z in the range y < z < x+y.
%C A370408 Another way to construct the sequence: Place 1's at the earliest permitted positions (in this case, at Fibonacci indices). Each subsequent value (2’s, 3’s, etc.) is placed at the earliest permitted indices not already occupied by a smaller value. For example, 3's could be placed in a Fibonacci pattern beginning with 7, 9 (7, 9, 16, 25, etc.), but i=7+9=16 is already occupied by the value 2, so 3 gets the next smallest position i=17. i=9+17=26 is again occupied by a 2, so we give 3 the next smallest unoccupied position i=27.
%H A370408 Michael S. Branicky, <a href="/A370408/b370408.txt">Table of n, a(n) for n = 1..10000</a>
%t A370408 list={1};Do[k=1;While[lst=Join[list,{k}];!And@@(And@@(({a,b,c}=#;(-a+b+c)(a-b+c)(a+b-c))<=0&/@Subsets[Flatten[Position[lst,#]],{3}])&/@Union@lst),k++];AppendTo[list,k],{n,92}];list (* _Giorgos Kalogeropoulos_, Feb 20 2024 *)
%o A370408 (Python)
%o A370408 from itertools import combinations as C, count, islice
%o A370408 def agen(): # generator of terms
%o A370408     yield from [1, 1, 1]
%o A370408     sides = {1: [1, 2, 3]}
%o A370408     for n in count(4):
%o A370408         an = next(an for an in count(1) if an not in sides or all(not all((n<b+c, b<n+c, c<n+b)) for b, c in C(sides[an], 2)))
%o A370408         yield an
%o A370408         if an not in sides: sides[an] = []
%o A370408         sides[an].append(n)
%o A370408 print(list(islice(agen(), 93))) # _Michael S. Branicky_, Feb 24 2024
%Y A370408 Cf. A367196, A107572 (triangle side lengths), A100480.
%K A370408 nonn
%O A370408 1,4
%A A370408 _Neal Gersh Tolunsky_, Feb 17 2024
%E A370408 More terms from _Giorgos Kalogeropoulos_, Feb 20 2024