ABSTRACT
When n = 1, take (1,1). When n = 4, take (1,1,3,4,2,3,2,4). When n = 5, take (2,4,2,3,5,4,3,1,1,5). When n > 5, use the construction (as ordered pairs):
n = 4s :
(4s + r − 1, 8s− r + 1) r = 1, . . . , 2s (r, 4s− r − 1) r = 1, . . . , s− 2 (s + r + 1, 3s− r) r = 1, . . . , s− 2 (s − 1, 3s), (s, s + 1), (2s, 4s− 1), (2s+ 1, 6s)
n = 4s+ 1 :
(4s + r + 1, 8s− r + 3) r = 1, . . . , 2s (r, 4s− r + 1) r = 1, . . . , s (s + r + 2, 3s− r + 1) r = 1, . . . , s− 2 (s + 1, s+ 2), (2s + 1, 6s+ 2), (2s+ 2, 4s+ 1)
2. (O’Keefe) A hooked Skolem sequence of order n exists if and only if n ≡ 2, 3 (mod 4). When n = 2, take (1,1,2,0,2). When n = 3, take (3,1,1,3,2,0,2). For n ≥ 6, use the construction (as ordered pairs):
n = 4s + 2 :
(r, 4s− r + 2) r = 1, . . . , 2s (4s + r + 3, 8s− r + 4) r = 1, . . . , s− 1 (5s + r + 2, 7s− r + 3) r = 1, . . . , s− 1 (2s + 1, 6s+ 2), (4s + 2, 6s+ 3), (4s + 3, 8s+ 5), (7s + 3, 7s+ 4)
n = 4s − 1 :
(4s + r, 8s− r − 2) r = 1, . . . , 2s− 2 (r, 4s− r − 1) r = 1, . . . , s− 2 (s + r + 1, 3s− r) r = 1, . . . , s− 2 (s − 1, 3s),(s, s+ 1),(2s, 4s− 1),(2s+ 1, 6s− 1),(4s, 8s− 1)
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).
