Extended near Skolem sequences Part I

A k ‐extended q ‐near Skolem sequence of order n , denoted by N n q ( k ) , is a sequence s 1 , s 2 , … , s 2 n − 1 , where s k = 0 and for each integer ℓ ∈ [ 1 , n ] \ { q } there are two indices i , j such that s i = s j = ℓ and ∣ i − j ∣ = ℓ . For a N n q ( k ) to exist it is necessary that q ≡ k(mod2 ) when n ≡ 0 , 1(mod4 ) and q ≢ k(mod2 ) when n ≡ 2 , 3(mod4 ) , and it is also necessary that ( n , q , k ) ≠ ( 3 , 2 , 3 ) , ( 4 , 2 , 4 ) . Any triple ( n , q , k ) satisfying these conditions is called admissible . In this article, which is part I of three articles, we construct sequences N n q ( k ) for all admissible ( n , q , k ) with 1 ≤ q ≤ ⌊n − 1 3 ⌋ and also for all admissible ( n , q , k ) with q ∈ [ ⌊n 3 ⌋ , n ] and k ∈ [ 1 , ⌊2 n 3 ⌋ − 1 ] .