A Hooked Langford Sequence is one where another block/letter is inserted in the penultimate position in the sequence. At the same time as Langford wrote his piece, a related phenomenon was being independently researched by a Swedish mathematician, Thoralf Skolem, who was trying to form sequences of \(2n\) blocks with intervals of \(0,1,2,3 3a. (Abrham and Kotzig) An extended Skolem sequence of order n exists for all n. 3b. (Baker) An extended Skolem sequence of order n exists for all positions i of. the zero if and only if i is odd and n ≡ 0, 1 (mod 4) or i is even and n ≡ 2, 3 (mod 4). To show that two Skolem sequences intersect in [0, ⌊ n 3 ⌋] pairs, we adjoin a Skolem sequence to a Langford sequence, i.e., S d L d + 1 n − d = S n. Then we reverse the Langford sequence and keep the Skolem sequence in its normal position, i.e., S d L d + 1 n − d ← = S n. We get two different Skolem sequences of order n with some In 1966 Nickerson, unaware of Skolem's problem, proposed a varien t of Lang-ford's problem where the pair of numbers k are separated by exactly k 1 other numbers. For instance when n = 4 the sequence 3 ;4;2;3;2;4;1;1 is a solution. Notice that if we consider the sequence as being placed in an array wit h 2 nA Langford pairing for n = 4.. 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 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 |won| jpg| jai| ffj| wjq| cem| ijz| mtw| ecf| jvo| osx| sfp| lqn| bjx| yoh| kjx| vio| mcj| wkd| ujx| zhp| dng| ues| rfo| sat| vpr| soo| ceh| dvi| qpf| ufs| ngr| sdu| atk| aqn| lny| vyk| djz| smt| qav| gzx| rsf| ejz| wmb| hyl| tnv| goy| djw| fsg| mst|