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Stolarsky-Harborth Constant


Let b(k) be the number of 1s in the binary expression of k, i.e., the binary digit count of 1, giving 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120) for k=1, 2, .... Then the number of odd binomial coefficients (k; j) where 0<=j<=k is 2^(b(k)) (Glaisher 1899, Fine 1947). This means that 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)

the first few terms of which are 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, ... (OEIS A006046).

The terms of this sequence are given by the recurrence

 f_n={3f_(n/2)   for n even; 2f_((n-1)/2)+f_((n+1)/2)   for n odd
(2)

with f_0=0 and f_1=1. The special case of n a power of 2 gives

 f(2^k)=3^k.
(3)
Stolarsky-HarborthConstant

Taking a cue from equation (3), the function f_n can be well approximated by n^theta, where

 theta=log_23=(ln3)/(ln2)=1.58496...
(4)

(OEIS A020857). In addition, f_n/n^theta oscillates between a minimum near 0.81... and a maximum at 1 in a fractal-like manner, as illustrated above. Stolarsky (1977) and Harborth (1977) studied the asymptotic behavior of f_n/n^theta. Define

alpha=lim sup_(n->infty)(f_n)/(n^theta)
(5)
beta=lim inf_(n->infty)(f_n)/(n^theta),
(6)

where lim inf is the infimum limit and lim sup is the supremum limit. Stolarsky (1977) showed that

 1<=alpha<=1.052 
0.72<=beta<=(9/7)(3/4)^theta<=0.815
(7)

and conjectured that

alpha=1
(8)
beta=(9/7)(3/4)^theta=3^theta/7=0.814931
(9)

(Harborth 1977, Stolarsky 1977). Harborth (1977) subsequently proved that alpha=1, but that the correct value for beta, called the Stolarsky-Harborth constant by Finch, is equal to beta=0.812556. A more exact value is

 beta=0.81255655901600638769...
(10)

(OEIS A077464).

Stolarsky-HarborthMinima

The value of this constant can be computed by examining the sequence

 q_r=(f_(n_r))/(n_r^theta),
(11)

where n_r is defined by n_0=1 and the recurrence

 n_r=2n_(r-1)+/-1,
(12)

with the sign chosen to minimize q_r. The resulting points (n_r,q_r) are local minima, illustrated above. The first few values of (n_r,q_r) are (1, 1), (3, 5), (5, 11), (11, 37), (21, 103), (43, 317), (87, 967), (173, 2869), ... (OEIS A077465 and A077466; Harborth 1977). The number of 1s in the binary representation of the minimal n_r are then 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, ... (OEIS A077467). Harborth (1977) computed beta to six digits using rigorous inequalities on q_(19), but opined "at the end, we remark that q from [q=lim_(r->infty)q_r] will probably be the exact value of beta."

Note that Harborth's recurrence does not necessarily give the cumulative minima, since it will miss a local minimum at n_r-2 if the value of the function is less when evaluated at n_r-1 than at n_r. The sequence giving all local minima is therefore 1, 3, 5, 11, 21, 43, 87, 171, 173, 347, 693, 1387, 2775, 5547, 5549, ... (OEIS A084230), where the "missing" terms 171, 5547, ... have been added back in.


See also

Batrachion, Binary, Binomial Coefficient, Blancmange Function, Digit Count, Hofstadter-Conway $10,000 Sequence, Pascal's Triangle, Rudin-Shapiro Sequence

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References

Finch, S. R. "Stolarsky-Harborth Constant." §2.16 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 145-151, 2003.Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer. Math. Monthly 54, 589-592, 1947.Glaisher, J. W. L. "On the Residue of a Binomial-Theorem Coefficient with Respect to a Prime Modulus." Quart. J. Math. 30, 150-156, 1899.Harborth, H. "Number of Odd Binomial Coefficients." Proc. Amer. Math. Soc. 62, 19-22, 1977.Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., p. 1, 1976.Kimball, S. H.; Hatcher, T. R.; Riley, J. A>; and Moser, M. "Solution to Problem E 1288: Odd Binomial Coefficients." Amer. Math. Monthly 65, 368-369, 1958.Lakhtakia, A. and Messier, R. "Self-Similar Sequences and Chaos from Gauss Sums." Computers and Graphics 13, 59-62, 1989.Lakhtakia, A.; Messier, R.; Varadan, V. K.; and Varadan, V. V. "Fractal Sequences Derived from the Self-Similar Extensions of the Sierpinski Gasket." J. Phys. A 21, 1925-1928, 1988.McIlroy, M. D. "The Number of 1's in Binary Integers: Bounds and Extremal Properties." SIAM J. Comput. 3, 255-261, 1974.Roberts, J. B. "On Binomial Coefficient Residues." Canad. J. Math. 9, 363-370, 1957.Sloane, N. J. A. Sequences A000120, A006046/M2445, A020857, A077464, A077465, A077466, A077467, and A084230 in "The On-Line Encyclopedia of Integer Sequences."Stolarsky, K. B. "Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity." SIAM J. Appl. Math. 32, 717-730, 1977.Wolfram, S. "Geometry of Binomial Coefficients." Amer. Math. Monthly 91, 566-571, 1984.

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Stolarsky-Harborth Constant

Cite this as:

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

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