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Stern's Diatomic Series

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SternsDiatomicSeries

Stern's diatomic series is the sequence whose terms beginning at index 1 are

1, 1,2, 1,3,2,3, 1,4,3,5,2,5,3,4,
(1)

... (OEIS A002487) which arises in the Calkin-Wilf tree. It is sometimes also known as the fusc function (Dijkstra 1982).

For integers n>=2, the nth term is given by the recurrence relation

a_n={a_(n/2) for n even; a_((n-1)/2)+a_((n+1)/2) for n odd,
(2)

with a_0=0 and a_1=1. For n>=1, a sum formula is given by

a_n=sum_(k=0)^(n-1)((k; n-k-1) (mod 2)).
(3)

The generating function G(x)=sum_(n=0)^(infty)a_nx^n, as a formal power series or for |x|<1, is given by

G(x)=xproduct_(k=0)^(infty)(1+x^(2^k)+x^(2^(k+1)))
(4)
=x+x^2+2x^3+x^4+3x^5+....
(5)

See also

Calkin-Wilf Tree, Stern-Brocot Tree

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References

Calkin, N. and Wilf, H. S. "Recounting the Rationals." Amer. Math. Monthly 107, 360-363, 2000. https://doi.org/10.1080/00029890.2000.12005205.Dijkstra, E. W. Selected Writings on Computing: A Personal Perspective. New York: Springer-Verlag, pp. 215-216 and 230-232, 1982.Gibbons, J.; Lester, D.; and Bird, R. "Functional Pearl: Enumerating the Rationals." J. Func. Prog. 16, 281-291, 2006. https://doi.org/10.1017/S0956796806005880.Sloane, N. J. A. Sequence A002487/M0141 in "The On-Line Encyclopedia of Integer Sequences."

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Stern's Diatomic Series

Cite this as:

Weisstein, Eric W. "Stern's Diatomic Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SternsDiatomicSeries.html

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