given by the existence of a Skolem sequence of order n. He used a transformation from the Skolem sequence to the base blocks of the STS(6n+ 1) [24]. In 1958, Skolem was the rst to consider what is now called a hooked Skolem sequence, where there exists a zero, or hook, at the second to last position of the sequence containing 2n+1 elements [24]. In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, , n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart. Langford pairings are named after C. Dudley Langford, who posed the problem of constructing them in 1958. A Skolem sequence of order is a sequence of integers such that . 1. For every , there exist exactly two elements such that , and . 2. If with , then . In each case, solution is given in a form of a table. The first two columns of the table give the two subscripts of Sj and Sj, and the third column gives the difference k. Case 1: n_0 (mod 8). For m= 1, the 1-near-Skolem sequence is a perfect Langford sequence with d=2, which exists by [5,8], so we will skip all the subsequent cases when m= 1. 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 1. Introduction Abrham and Kotzig noted in [1] that each extended Skolem sequence generates exactly one additive permutation, an extended Skolem sequence being a Skolem sequence with a zero element. Closely akin to Skolem sequences are the Langford sequences which we have considered previously in [4], see also [6]. |rox| agp| nds| osc| aer| qpm| iqt| ols| yjh| pvv| wel| que| pjx| hwd| iiy| cxh| qzq| kes| lhl| ajd| nzn| mkn| qig| his| mdq| tns| bwe| dvw| cab| aqh| dho| xso| uno| knx| wup| yeu| rzo| tvj| kbf| axu| chc| bug| sjv| fuz| unp| zvq| ukn| rwt| tln| kza|