Bounds on Number of Odd Terms in Pascal's Triangle

Theorem

Let $P_n$ be the number of odd elements in the first $n$ rows of Pascal's triangle.

Then:

$0 \cdotp 812 \ldots < \dfrac {P_n} {n^{\lg 3} } < 1$

where $\lg 3$ denotes logarithm base $2$ of $3$.

The lower bound $0 \cdotp 812 \ldots$ is known as the Stolarsky-Harborth constant.

Proof

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Also see

• Definition:Stolarsky-Harborth Constant

Sources

• 1977: Heiko Harborth: Number of Odd Binomial Coefficients (Proc. Amer. Math. Soc. Vol. 62: pp. 19 – 22)  www.jstor.org/stable/2041936

• 1977: Kenneth B. Stolarsky: Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity (SIAM J. Appl. Math. Vol. 32: pp. 717 – 730)  www.jstor.org/stable/2100181

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,81256 6 \ldots$

• Weisstein, Eric W. "Stolarsky-Harborth Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Stolarsky-HarborthConstant.html