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Gould's sequence

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The self-similar sawtooth shape of Gould's sequence

Gould's sequence is an integer sequence consisting only of powers of two, that counts the odd numbers in each row of Pascal's triangle. It begins:[1][2]

1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, ... (sequence A001316 in the OEIS)

For instance, there is one odd number in the first row of Pascal's triangle, and two (both 1) in the second and third rows, but the fourth row (1, 3, 3, 1) has four odd numbers.

The sequence is named after Henry W. Gould.

Additional interpretations

The nth value in the sequence (starting from n = 0) gives the highest power of two that divides the central binomial coefficient , the numerator of (expressed as a fraction in lowest terms).[1]

Sierpinski triangle generated by Rule 90. Gould's sequence counts the number of live cells in each row of this pattern.

Gould's sequence also gives the number of live cells in the nth generation of the Rule 90 cellular automaton starting from a single live cell.[1] It has a characteristic exponentially growing sawtooth shape that can be used to recognize physical processes that behave similarly to Rule 90.[3]

The exponents in the powers of two of Gould's sequence themselves form an integer sequence,

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, ... (sequence A000120 in the OEIS)

in which the nth value gives the number of nonzero bits in the binary representation of the number n.[1][2] Taking this sequence modulo two gives the Thue–Morse sequence.[4]

Recursive construction and self-similarity

The first 2i values in the sequence may be constructed by recursively constructing the first 2i − 1 values, and then concatenating the doubles of the first 2i − 1 values. For instance, concatenating the first four values 1, 2, 2, 4 with their doubles 2, 4, 4, 8 produces the first eight values. Because of this doubling construction, the first occurrence of each power of two 2i in this sequence is at position 2i − 1.[1]

Gould's sequence, the sequence of its exponents, and the Thue–Morse sequence are all self-similar: they have the property that the subsequence of values at even positions in the whole sequence equals the original sequence, a property they also share with some other sequences such as Stern's diatomic sequence.[5][6] In Gould's sequence, the values at odd positions are double their predecessors, while in the sequence of exponents, the values at odd positions are one plus their predecessors.

References

  1. ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A001316 (Gould's sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ a b Pólya, George; Tarjan, Robert E.; Woods, Donald R. (2009), Notes on Introductory Combinatorics, Progress in Computer Science and Applied Logic, vol. 4, Springer, p. 21, ISBN 9780817649531.
  3. ^ Claussen, Jens Christian; Nagler, Jan; Schuster, Heinz Georg (2004), "Sierpinski signal generates 1∕f α spectra", Physical Review E, 70: 032101, arXiv:cond-mat/0308277, Bibcode:2004PhRvE..70c2101C, doi:10.1103/PhysRevE.70.032101.
  4. ^ Northshield, Sam (2010), "Sums across Pascal's triangle mod 2", Congressus Numerantium, 200: 35–52, MR 2597704.
  5. ^ Gilleland, Michael, Some Self-Similar Integer Sequences, OEIS, retrieved 2016-09-10.
  6. ^ Schroeder, Manfred (1996), "Fractals in Music", in Pickover, Clifford A. (ed.), Fractal Horizons, New York: St. Martin's Press, pp. 207–223 {{citation}}: Unknown parameter |editorlink= ignored (|editor-link= suggested) (help). As cited by Gilleland.
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