A119632 Lengths of successive runs in A160357, where a run here means a string of alternating terms.

1, 1, 3, 1, 11, 1, 4, 10, 1, 4, 28, 1, 10, 24, 1, 8, 1, 2, 1, 1, 4, 1, 9, 4, 1, 2, 36, 1, 12, 4, 1, 2, 1, 3, 28, 1, 10, 52, 1, 18, 1, 32, 1, 12, 15, 38, 1, 14, 32, 1, 12, 1, 44, 1, 16, 1, 148, 1, 50, 7, 22, 1, 8, 3, 4, 1, 2, 70, 1, 24, 1, 114, 1, 42, 1, 200, 1, 68, 6, 1, 2, 13

Offset: 1

Author

N. J. A. Sloane and Allan Wilks, Jun 10 2006

Keywords

• nonn

Comments

Gives a highly compressed version of A005132.

The encoding of Recamán's sequence a(n) = A005132 using A119632 is easy - A119632 counts runs of alternating i(n)'s, where i(n) = (a(n)-a(n-1))/n = A160357(n).

Note that i(n) is always +1 or -1. Each run ends when i(n) = i(n+1).

Here is pseudo-code to reconstruct Recamán's sequence from A119632, which we will call I(n):

a(0) = 0

n = 1

i = 1

for k = 1..oo {

for j = 1..I(k) {

a(n) = a(n-1) + n*i

n = n+1

i = -i

}

i = -i

}

The gzipped file attached to A119632 represents the first 1470117206801829 terms of A005132. The longest run of alternating i(n)'s (maximal value found so far in A119632) is 232144588914. There are 64094657 runs encoded in the gzipped file.

Examples

			A160357 begins 1, 1; 1; -1, 1, 1; 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1; 1; -1, 1, -1, -1; 1, -1, 1, -1, 1, -1, 1, -1, 1, 1; 1; ..., where semicolons demark the successive runs.
		

Links

• Allan Wilks, Table of n, a(n) for n = 1..100000
• Allan Wilks, The first 64094657 terms (gzipped). (A large file. This encodes the first 1470117206801829 terms of A005132!)
• Index entries for sequences related to Recamán's sequence

Crossrefs

Cf. A005132, A160357.

Extensions

Entry expanded by N. J. A. Sloane, Jul 15 2011