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A309503 Iteratively replace the sum of two sequentially chosen consecutive integers with those integers.

%I A309503 #21 Apr 02 2020 00:42:26
%S A309503 1,2,1,2,4,5,2,4,5,2,8,4,5,2,8,11,8,4,13,14,15,16,4,13,4,14,8,11,16,4,
%T A309503 8,13,22,23,24,4,8,13,4,14,8,13,14,24,4,14,15,30,15,16,24,8,33,34,13,
%U A309503 22,36,37,14,24,8,13,4,14,24,16,41,42,8,13,22,44
%N A309503 Iteratively replace the sum of two sequentially chosen consecutive integers with those integers.
%F A309503 To generate the sequence, start with the integers, A_0={1,2,3,4,5,...}. To generate A_{n+1} calculate x = A_n(n) + A_n(n+1). Replace the next instance of x in A_n (after A_n(n+1)) with A_n(n), A_n(n+1). The limit of this process gives the sequence.
%e A309503 A_0 = {1,2,3,4,5,...}.
%e A309503 A_0(0) + A_0(1) = 1 + 2 = 3, which is found after A_0(1), so A_1 = {1,2,1,2,4,5,...}.
%e A309503 A_1(1) + A_1(2) = 2 + 1 = 3, which is *not* found after A_1(2), so A_2 = A_1.
%e A309503 A_2(2) + A_2(3) = 1 + 2 = 3, A_3 = A_2.
%e A309503 A_3(3) + A_3(4) = 2 + 4 = 6, A_4 = {1,2,1,2,4,5,2,4,7,8,9,10,...}.
%e A309503 A_4(4) + A_4(5) = 4 + 5 = 9, A_5 = {1,2,1,2,4,5,2,4,7,8,4,5,10,...}.
%t A309503 T = Range[100]; Do[s = T[[i]] + T[[i + 1]]; Do[If[T[[j]] == s, T = Join[ T[[;; j-1]], {T[[i]], T[[i+1]]}, T[[j+1 ;;]]]; Break[]], {j, i+2, Length@ T}], {i, Length@T}]; T (* _Giovanni Resta_, Sep 20 2019 *)
%o A309503 (Kotlin)
%o A309503 fun generate(len: Int): List<Int> {
%o A309503     fun gen_inner(len: Int, level: Int): List<Int> {
%o A309503         if (level < 1) return (1..len).toList()
%o A309503         val prev = gen_inner(len, level - 1)
%o A309503         if (level == len) return prev.take(len)
%o A309503         val (a, b) = prev[level - 1] to prev[level]
%o A309503         return if (prev.drop(level + 1).contains(a+b)) {
%o A309503             prev.indexOfFirst { it == a+b }.let { idx ->
%o A309503                 prev.take(idx) + a + b + prev.drop(idx + 1)
%o A309503             }
%o A309503         } else prev
%o A309503     }
%o A309503     return gen_inner(len, len)
%o A309503 }
%o A309503 (PARI)
%o A309503 a = vector(92, k, k);
%o A309503 for (n=1, #a-1, s=a[n]+a[n+1]; print1 (a[n] ", "); for (k=n+2, #a - 1, if (a[k]==s, a=concat([a[1..k-1], a[n..n+1], a[k+1..#a]]); break)))
%Y A309503 This sequence is similar to A309435 except it uses addition instead of multiplication.
%K A309503 nonn,easy
%O A309503 1,2
%A A309503 _Matthew Malone_, Aug 05 2019