Stolarsky-Harborth Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $b(k)$ be the number of 1s in the Binary expression of $k$. Then the number of Odd Binomial Coefficients ${k\choose j}$ where $0\leq j\leq k$ is $2^{b(k)}$ (Glaisher 1899, Fine 1947). The number of Odd elements in the first $n$ rows of Pascal's Triangle is

$$f(n)=\sum_{k=0}^{n-1} 2^{b(k)}.$$ (1)

This function is well approximated by $n^\theta$, where

$$\theta\equiv{\ln 3\over\ln 2}=1.58496\ldots.$$ (2)

Stolarsky and Harborth showed that

$$0.812556\leq \liminf_{n\to\infty} {f(n)\over n^\theta} < 0.812557 < \limsup_{n\to\infty} {f(n)\over n^\theta}=1.$$ (3)

The value

$$SH=\liminf_{n\to\infty} {f(n)\over n^\theta}$$ (4)

is called the Stolarsky-Harborth constant.

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/stlrsky/stlrsky.html

Fine, N. J. ``Binomial Coefficients Modulo a Prime.'' Amer. Math. Monthly 54, 589-592, 1947.

Wolfram, S. ``Geometry of Binomial Coefficients.'' Amer. Math. Monthly 91, 566-571, 1984.