  We compute the number of solutions to the Skolem pairings problem, S(n), and
to the Langford variant of the problem, L(n). These numbers correspond to the
sequences A059106, and A014552 in Sloane's Online Encyclopedia of Integer
Sequences. The exact value of these numbers were known for any positive integer
n < 24 for the first sequence and for any positive integer n < 27 for the
second sequence. Our first contribution is computing the exact number of
solutions for both sequences for any n < 30. Particularly, we report that S(24)
= 102, 388, 058, 845, 620, 672. S(25) = 1, 317, 281, 759, 888, 482, 688. S(28)
= 3, 532, 373, 626, 038, 214, 732, 032. S(29) = 52, 717, 585, 747, 603, 598,
276, 736. L(27) = 111, 683, 611, 098, 764, 903, 232. L(28) = 1, 607, 383, 260,
609, 382, 393, 152. Next we present a parallel tempering algorithm for
approximately counting the number of pairings. We show that the error is less
than one percent for known exact numbers, and obtain approximate values for
S(32) ~ 2.2x10^26 , S(33) ~ 3.6x10^27, L(31) ~ 5.3x10^24, and L(32) ~ 8.8x10^25
