bibkey: yanev2017a030101
authors: David W. Wilson; Ralf Stephan; Velin Yanev; Alois P. Heinz; Henry Bottomley
year: 2017
title: "OEIS A030101, binary digit reversal, with Yanev's position identity"
doi: null
url: https://oeis.org/A030101
claim: "A030101 %N: a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number. A030101 %F: Conjecture: a(n) = 2*w(n) - 2*w(A053645(n)) - 1 for n > 0, where w = A264596. - _Velin Yanev_, Sep 12 2017"
strata_touched:
  - D5/S1/Digit/YanevBinaryReversalPositionIdentity
license: citation-only
triage: anchor

OEIS A030101: binary digit reversal

The entry defines A030101 by reversal of the canonical binary digits. Its Stephan recurrence supplies the formal definition of rev. The referenced sequence A264596 supplies Heinz’s recurrence for w, and A053645 removes the largest binary power from a positive input.

Verified locator

• URL: https://oeis.org/A030101
• A030101 %N (verbatim): a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number.
• A030101 conjecture %F (verbatim): Conjecture: a(n) = 2w(n) - 2w(A053645(n)) - 1 for n > 0, where w = A264596. - Velin Yanev, Sep 12 2017
• A030101 recurrence %F (verbatim): a(n) = 0, a(2n) = a(n), a(2n+1) = a(n) + 2^(floor(log_2(n)) + 1). For n > 0, a(n) = 2*A030109(n) - 1. - Ralf Stephan, Sep 15 2003
• A030101 %A (verbatim): David W. Wilson
• A264596 %N (verbatim): Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.
• A264596 recurrence %F (verbatim): a(2n) = a(n), a(2n+1) = a(n) + n+1, a(0) = 0. - Alois P. Heinz, Nov 19 2015
• A053645 %N (verbatim): Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.
• A053645 %F (verbatim): a(n) = n - 2^A000523(n).

The locator and quotations were read from the three OEIS text endpoints on 2026-09-16. The A030101 recurrence begins with the printed text a(n) = 0. The data line begins at zero, and the formal total function uses that intended zero boundary. No priority claim beyond the cited entries is made.