Wrecker ball sequence

This page is related to the so-called "wrecker ball sequences" defined by x(n) = x(n-1) - e*n, n >= 1, where e = sign(x(n-1)) if the corresponding x(n) does not occur earlier as x(k), k < n, or -sign(x(n-1)) else. Depending on the starting value x(0), we consider:

• the "orbit" of x(0) = n: A248939 (rows = the full sequences x(k)), stopping at 0.
• the length of the orbits: A228474 (lists the lengths - 1 = number of steps to reach 0).
• minimal value of the orbit of x(0) = n: A248952,
• maximal value of the orbit of x(0) = n: A248953,
• the sum of all x(k): A248961 (row sums of A248939),
• corresponding partial sums: A248973.

Introduction

The basic idea is very similar to that of the celebrated Recamán sequence A005132, but the behaviour of the sequence is completely different.

Examples

Some sequences list the orbit of given starting values, separately from the table A248939 which contains all of them it its rows:

• A248940 (row 7),
• A248941 (row 17),
• A248942 (row 20).

Theoretical results

Triangular numbers

It is obvious that for triangular numbers n = A000217(k) = 1 + 2 + ... + k = k(k+1)/2, the sequence starting with x(0) = n goes very simply "straight towards zero":

• A248939(n,.) = (n, n-1, n-1-2, n-1-2-3, ..., n - A000217(j), ..., 0).
• A228474(n) = k (number of steps to reach zero).
• A248952(n) = 0 (minimal value), A248953(n) = k (maximal value).

There are the "minimal" values of A228474: all subsequent values will be larger than this one, A228474(m) > A228474(n) for all m > n, and the only smaller values A228474(m) are those where m = A000217(j) for some j < k.

Parity

Robert Gerbicz has shown that floor((A228474(n)-1)/2)+n is always odd.

[TO_DO: Put illustration and proof here]

Clusters

There are many "clusters" of consecutive values n..m for which the ranges A248953(n..m) and/or A228474(n..m) comprises only values lying very closely and regularly together; in particular:

• A248953(n..m) = n..m for n..m = {0..3, 14..15, 26..28, 35..36, 41..43, ...}, but also the less trivial A248953(12..13) = 365..366 and A248953(31..33) = 3702823..3702825 etc.,
• A228474(n..m) is a sequence of values with first differences = +-2: e.g., A228474(4..5) = (24, 26), A228474(8..9) = 12..14, A228474(12..13) = (123, 125), A228474(18..19) = (20, 22), A228474(23..24) = (29, 31), A228474(26) = (23, 25), A228474(31..34) = (532009, 532007, 532009, 532011), A228474(38..39) = (213355, 213353), A228474(41..43) = (33, 31, 33), A228474(47..48) = (110, 112), A228474(49..50) = (82, 84), ...

Authoring information

This page has been created by M. F. Hasler, 09:58, 1 April 2019 (EDT)

Cite this page as

M. F. Hasler, Wrecker ball sequence.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Wrecker_ball_sequence]