%I #21 Apr 02 2020 00:42:26

%S 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 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 22,36,37,14,24,8,13,4,14,24,16,41,42,8,13,22,44

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

%F 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 A_0 = {1,2,3,4,5,...}.

%e 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 A_1(1) + A_1(2) = 2 + 1 = 3, which is *not* found after A_1(2), so A_2 = A_1.

%e A_2(2) + A_2(3) = 1 + 2 = 3, A_3 = A_2.

%e A_3(3) + A_3(4) = 2 + 4 = 6, A_4 = {1,2,1,2,4,5,2,4,7,8,9,10,...}.

%e 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 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 (Kotlin)

%o fun generate(len: Int): List<Int> {

%o fun gen_inner(len: Int, level: Int): List<Int> {

%o if (level < 1) return (1..len).toList()

%o val prev = gen_inner(len, level - 1)

%o if (level == len) return prev.take(len)

%o val (a, b) = prev[level - 1] to prev[level]

%o return if (prev.drop(level + 1).contains(a+b)) {

%o prev.indexOfFirst { it == a+b }.let { idx ->

%o prev.take(idx) + a + b + prev.drop(idx + 1)

%o }

%o } else prev

%o }

%o return gen_inner(len, len)

%o }

%o (PARI)

%o a = vector(92, k, k);

%o 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 This sequence is similar to A309435 except it uses addition instead of multiplication.

%K nonn,easy

%O 1,2

%A _Matthew Malone_, Aug 05 2019