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Wolfram Sequences


WolframSequence32

Wolfram (2002, p. 123) considered the sequence related to the Collatz problem obtained by iterating

 w_n={3/2w_(n-1)   for w_(n-1) even; 3/2(w_(n-1)+1)   for w_(n-1) odd
(1)

starting with w_1=1. This gives the sequence 1, 3, 6, 9, 15, 24, 36, 54, 81, 123, ... (OEIS A070885). The first 40 iterations are illustrated above, with each row being one iteration and the number obtained in that iteration represented in binary.

WolframSequence52

Another set of sequences are given by

 w_n={5/2w_(n-1)   for w_(n-1) even; 1/2(w_(n-1)+1)   for w_(n-1) odd
(2)

starting with various initial values w_1=n. Interestingly, while taking n=1, 2, 3, 4, 5, 7, 9, 10, ... give simple periodic sequences, the cases n=6, 8, give complicated aperiodic sequences. 100 iterations starting at each of w_1=1 to 10 are illustrated above.

Wolfram also considered the sequence 1, 1, 3, 3, 3, 5, 3, ... (OEIS A070864) defined by f(1)=f(2)=1 and

 f(n)=2+f(n-f(n-1))
(3)

(Wolfram 2002, p. 129, (b)), the sequence 1, 1, 2, 2, 2, 4, 3, 4, 4, 4, ... (OEIS A070867) defined by f(1)=f(2)=1 and

 f(n)=f(n-f(n-1)-1)+f(n-f(n-2)-1)
(4)

(Wolfram 2002, p. 129, (f)), and the sequence 1, 1, 2, 2, 2, 3, 3, 4, 3, 4, ... (OEIS A070868) defined by f(1)=f(2)=1 and

 f(f(n-1))+f(n-2f(n-1)+1)
(5)

(Wolfram 2002, p. 129, (h)).


See also

Collatz Problem

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References

Sloane, N. J. A. Sequences A070864, A070867, A070868, and A070885 in "The On-Line Encyclopedia of Integer Sequences."Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 129, 2002.

Referenced on Wolfram|Alpha

Wolfram Sequences

Cite this as:

Weisstein, Eric W. "Wolfram Sequences." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WolframSequences.html

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