Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,…

Abstract Stern's diatomic sequence appeared in print in 1858 and has been the subject of numerous papers since. Our goal is to present many of these properties, both old and new. We present a large set of references and, for many properties, we supply simple proofs or ones that complement existing proofs. Among the topics covered are what these numbers count (hyperbinary representations) and the sequence's surprising parallels with the Fibonacci numbers. Quotients of consecutive terms lead to an enumeration of the rationals. Other quotients lead to a map from dyadic rationals to the rationals whose completion is the inverse of Minkowski's? function. Along the way, we get a distant view of fractals and the Riemann hypothesis as well as a foray into random walks on graphs in the hyperbolic plane.

[1]  S. Vajda.  Fibonacci and Lucas Numbers and the Golden Section , 1989 .

[2]  M. Stern.  Ueber eine zahlentheoretische Funktion. , 1858 .

[3]  E. Kummer.  Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. , 1852 .

[4]  L. Carlitz.  A problem in partitions related to the Stirling numbers , 1964 .

[5]  Stefan Erickson,et al.  Apollonian Circle Packings , 2022 .

[6]  Marjorie Bicknell-Johnson,et al.  The Fibonacci Diatomic Array Applied to Fibonacci Representations , 2004 .

[7]  Sam Northshield,et al.  Sums across Pascal's triangle modulo 2 , 2010 .

[8]  Herbert S. Wilf,et al.  Recounting the Rationals , 2000, Am. Math. Mon..

[9]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers with Applications: Koshy/Fibonacci , 2001 .

[10]  C. L. Mallows,et al.  Apollonian circle packings:number theory , 2000 .

[11]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[12]  Jeffrey Shallit,et al.  Formal languages and number theory , 2000, Unusual Applications of Number Theory.

[13]  Edsger W. Dijkstra,et al.  Selected Writings on Computing: A personal Perspective , 1982, Texts and Monographs in Computer Science.

[14]  E. Kummer,et al.  Ueber die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. , 1852 .

[15]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[16]  I. Urbiha.  Some properties of a function studied by De Rham, Carlitz and Dijkstra and its relation to the (Eisenstein-)Stern's diatomic sequence , 2001 .

[17]  Jeffrey Shallit,et al.  The ring of k-regular sequences, II , 2003, Theor. Comput. Sci..

[18]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers With Applications , 2018 .

[19]  J. Conway.  On Numbers and Games , 1976 .

[20]  Douglas Lind,et al.  An extension of Stern’s diatomic series , 1969 .

[21]  J. Paradís,et al.  The Derivative of Minkowski's ?(x) Function , 2001 .

[22]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[23]  V. Hoggatt.  Fibonacci and Lucas Numbers , 2020, Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science.

[24]  Heiko Harborth.  NUMBER OF ODD BINOMIAL COEFFICIENTS , 1977 .