The Stolarsky-Harborth constant is the infimum limit of the number of odd entries in the first 
rows of Pascal's triangle
divided by 
.
Let 
be the number of 1s in the binary expression of 
, i.e., the binary digit count
of 1, giving 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, ... (OEIS A000120)
for 
,
2, .... Then the number of odd binomial
coefficients 
where 
is 
(Glaisher 1899, Fine 1947). This means that the number
of odd elements in the first 
rows of Pascal's triangle
is
 |
(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, for integers 
,
by the recurrence relation
 |
(2)
|
with 
and 
.
The special case of 
a power of 2 gives
 |
(3)
|
for integers 
.
Taking a cue from equation (3), the function 
can be well approximated by 
, where
 |
(4)
|
(OEIS A020857). In addition, 
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 
. Define
 |  |  |
(5)
|
 |  |  |
(6)
|
where lim inf is the infimum limit and lim sup is the supremum limit. Stolarsky (1977) showed that
 |
(7)
|
and conjectured that
 |  |  |
(8)
|
 |  |  |
(9)
|
(Harborth 1977, Stolarsky 1977). Harborth (1977, pp. 19-22) subsequently proved that 
and refuted the conjectured value of 
, called the Stolarsky-Harborth constant by Finch (2003),
obtaining 
.
A reported numerical value is
 |
(10)
|
(OEIS A077464).
Harborth (1977, pp. 20-22) obtained a rigorous six-decimal estimate by examining the sequence
 |
(11)
|
where 
is defined by 
and the recurrence
relation
 |
(12)
|
with the sign chosen to minimize 
. The resulting points 
are local minima,
illustrated above. The first few values of 
are (1, 1), (3, 5), (5, 11), (11, 37), (21, 103),
(43, 317), (87, 967), (173, 2869), ... (OEIS A077465
and A077466; Harborth 1977). The numbers of
1s in the binary representations of the minimal 
are then 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, ... (OEIS A077467).
Harborth (1977) computed 
to six digits using rigorous inequalities on 
, but opined "at the end, we remark that 
from [
] will probably be the exact value of
."
Harborth's recurrence relation can miss successive record minima, including 171 and 5547. A more complete sequence of record minima begins 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.
