A119632 - OEIS

%I #39 Jan 25 2025 02:12:11

%S 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,

%T 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,

%U 50,7,22,1,8,3,4,1,2,70,1,24,1,114,1,42,1,200,1,68,6,1,2,13

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

%C Gives a highly compressed version of A005132.

%C 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).

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

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

%C         a(0) = 0

%C         n = 1

%C         i = 1

%C         for k = 1..oo {

%C                 for j = 1..I(k) {

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

%C                         n = n+1

%C                         i = -i

%C                 }

%C                 i = -i

%C         }

%C 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.

%H Allan Wilks, <a href="/A119632/b119632.txt">Table of n, a(n) for n = 1..100000</a>

%H Allan Wilks, <a href="/A119632/a119632.gz">The first 64094657 terms (gzipped)</a>. (A large file. This encodes the first 1470117206801829 terms of A005132!)

%H <a href="/index/Rea#Recaman">Index entries for sequences related to Recamán's sequence</a>

%e 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.

%Y Cf. A005132, A160357.

%K nonn,changed

%O 1,3

%A _N. J. A. Sloane_ and _Allan Wilks_, Jun 10 2006

%E Entry expanded by _N. J. A. Sloane_, Jul 15 2011