1,1

The ideal substring length is related to A007501. It can be shown these are equivalent problems.

For n = 1, the ideal substring length is 2.

For n > 1:

n = 2, term 0 of A007501, substring length = 2

n = 3, term 1 of A007501, substring length = 3

n = 4, term 2 of A007501, substring length = 6

etc.

This has applications in text processing operations in computer languages where recursions or loops may not be possible (e.g. standard SQL). To remove extra spaces, one might be tempted to nest several replace operations but use the same substring length, or perhaps double or halve at each step, both of which will not clear as effectively as using substring lengths as indicated in A007501.

Given substring length p as indicated in A007501, sequence is p(p+1)-2.

To illustrate, suppose we have a string of repeating Xs.

n = 1: replace(string, "XX", "X"), the longest string which will reduce to "X" is "XX"

n = 2: replace(replace(string, "XX", "X"), "XX", "X") will reduce up to 4 Xs to "X"

n = 3: replace(replace(replace(string, "XXX", "X"), "XX", "X"), "XX", "X") up to 10 Xs

n = 4: replace(replace(replace(replace(string, "XXXXXX", "X"), "XXX", "X"), "XX", "X"), "XX", "X") up to 40 Xs

etc.

(Other) // q is this sequence, p is A007501

set q = 2

output q

repeat

set p = q / 2 + 1

set q = p * (p + 1) - 2

output q

end repeat

2, followed by A007501

Sequence in context: A086852 A084737 A153757 this_sequence A013549 A125805 A028404

Adjacent sequences: A159857 A159858 A159859 this_sequence A159861 A159862 A159863

easy,nonn

Russell Harper (russell.harper(AT)springboardnetworks.com), Apr 24 2009