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

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

Offset

1,2

Links

Table of n, a(n) for n=1..72.

Formula

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.

Example

A_0 = {1,2,3,4,5,...}.
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,...}.
A_1(1) + A_1(2) = 2 + 1 = 3, which is *not* found after A_1(2), so A_2 = A_1.
A_2(2) + A_2(3) = 1 + 2 = 3, A_3 = A_2.
A_3(3) + A_3(4) = 2 + 4 = 6, A_4 = {1,2,1,2,4,5,2,4,7,8,9,10,...}.
A_4(4) + A_4(5) = 4 + 5 = 9, A_5 = {1,2,1,2,4,5,2,4,7,8,4,5,10,...}.

Mathematica

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 *)

Prog

(Kotlin)
fun generate(len: Int): List<Int> {
    fun gen_inner(len: Int, level: Int): List<Int> {
        if (level < 1) return (1..len).toList()
        val prev = gen_inner(len, level - 1)
        if (level == len) return prev.take(len)
        val (a, b) = prev[level - 1] to prev[level]
        return if (prev.drop(level + 1).contains(a+b)) {
            prev.indexOfFirst { it == a+b }.let { idx ->
                prev.take(idx) + a + b + prev.drop(idx + 1)
            }
        } else prev
    }
    return gen_inner(len, len)
}
(PARI)
a = vector(92, k, k);
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)))

Crossrefs

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

Sequence in context: A072727 A292601 A301364 * A057061 A307729 A245660

Adjacent sequences: A309500 A309501 A309502 * A309504 A309505 A309506

Keyword

nonn,easy

Author

Matthew Malone, Aug 05 2019

Status

approved