Counting Skolem Sequences

Abstract: 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