bibkey: adamson2023a002487 authors: N. J. A. Sloane; Gary W. Adamson year: 2023 title: “OEIS A002487, Stern’s diatomic series, with the Adamson Farey-tree inverse conjecture” doi: null url: https://oeis.org/A002487 claim: “%N Stern’s diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2n) = a(n), a(2n+1) = a(n) + a(n+1). %C It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7.” strata_touched:

• D5/S3/PrimeForms/SternBrocot/AdamsonSternFareyInverse license: citation-only triage: anchor

Stern’s diatomic series and Adamson’s Farey-tree inverse relation

The year 2023 records Gary W. Adamson’s December 18, 2023 comment. The OEIS AUTHOR line credits N. J. A. Sloane. The objects are the Stern sequence A002487 and the numerator and denominator sequences A007305 and A007306.

The Farey-tree listing starts with 0/1 and 1/1 at indices 0 and 1. Each subsequent level lists only the new mediants, in increasing order in [0,1]. Writing r = 2^m + j with 1 <= j <= 2^m, entry r is position 2(j-1)+1 of full mediant row m+1. Thus entry 13 is 4/7, while stern(12)=2.

Only Adamson’s modular-inverse sentence is settled for positive natural indices. Yurramendi’s frequency conjecture, Torres’s conjectures and other Farey-tree properties are not claimed.

Verified locator

The locator readings are dated 2026-09-15.

• URL: https://oeis.org/A002487
• A002487 %N (verbatim): Stern’s diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2n) = a(n), a(2n+1) = a(n) + a(n+1).
• A002487 %C, Gary W. Adamson, Dec 18 2023 (verbatim): It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7.
• A002487 %A (verbatim): N. J. A. Sloane
• A007305 URL: https://oeis.org/A007305
• A007305 %N (verbatim): Numerators of Farey (or Stern-Brocot) tree fractions.
• A007306 URL: https://oeis.org/A007306
• A007306 %N (verbatim): Denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range [0,1]).

Prior literature

Bates, Bunder and Tognetti, “Linking the Calkin–Wilf and Stern–Brocot trees”, European Journal of Combinatorics 31 (2010), doi:10.1016/j.ejc.2010.04.002, gives a transposition relation between the two trees and does not state this modular-inverse claim.

Yurramendi’s 2014 %F identity on A002487, a(k+1)*a(2^m+k) - a(k)*a(2^m+(k+1)) = 1, is stated there without proof; the module proves its additive natural-number form for k < 2^m as the private theorem stern_determinant.