Duffin's Sequence for Pi

Duffin's sequence for positive number is defined by:
,
where is the integer part of (or the floor of ). For a given , is a sequence of digits 0, 1, 2, ..., 6.
If , we get 0123456012345601234560123456…. Does this sequence have period 0123456?

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References
[1] R. J. Duffin, "The Patron Saint of Mathematics," The Mathematical Intelligencer, 15(1), 1993 p. 52. doi:10.1007/BF03025258.
[2] I. Vidav, On Division with Remainder and More (in Slovenian), Ljubljana: DMFA, 2016 pp. 85–86.
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