bibkey: layman2011a193020
authors: John W. Layman
year: 2011
title: OEIS A193020, distinct binary self-shuffles
doi: null
url: https://oeis.org/A193020
claim: "For positive n, A193020 defines a(n) as the number of distinct shuffles of the binary expansion of n with itself. Revision 13 states separate eventual assertions including a(2^k+1)=k^2 and a(2^k-4)=k*(k-1)/2, each for sufficiently large k, with the literal lower-bound caveat."
strata_touched:
  - D5/S1/Words/EndpointShuffleSupport
  - D5/S1/Words/DoubleTailShuffleSupport
license: CC-BY-SA-4.0
triage: anchor

OEIS A193020

For positive n, the published sequence a(n) counts the distinct output words obtained by order-preserving interleavings of the ordinary binary expansion of n with itself; different interleaving schedules that give the same word are counted once. The formal definitions are total on natural inputs: binaryWord uses the empty list at n=0, and distinctSelfShuffleCount counts its distinct self-shuffles. The OEIS entry itself includes a(0). Only the selected eventual events here use positive inputs; the formal empty-list convention at zero is not needed for either selected event and is not asserted to be the source’s convention.

The separate source assertions used here are the eventual identities a(2^k+1)=k^2 and a(2^k-4)=k*(k-1)/2, each with the source’s sufficiently-large lower-bound caveat. Each remains a distinct assertion. The quotient in the minus-four identity is natural-number division.

Verified locator

John W. Layman (2011), OEIS A193020, revision 13. Stable entry URL: https://oeis.org/A193020.

The OEIS content is licensed CC-BY-SA-4.0.

Scope of the source and proof

The count definition and the two separately selected eventual assertions are attributed to John W. Layman and the OEIS contributors. The support proofs are repository results, with acknowledgement of that source, rather than proofs supplied by the entry. The lower-bound caveat is retained; the repository proofs supply K=1 for plus one and K=3 for minus four.

The bounded literature search identified no exact whole-answer hit in its inspected scope. This is not a certificate of worldwide openness, absence of prior proofs or priority. Restivo’s full chapter, the Li-Qin and Guo-Xie journal bodies, uninspected Datko-Pawlik tables and displayed lists, remaining citation branches, and broader or unindexed work were not read. Weighted shuffle coefficients count multiplicities and do not by themselves supply the distinct-support argument used here. No other A193020 identity is resolved by these two claims.