Enumeration of combinations with restricted differences and cospan

Abstract For the p -combination 1≤ x 1 x 2 x p ≤ n the differences are d j =x j+1 −x j , j =1(1) p −1, the span is d=x p −x 1 , and the cospan is n−d . An explicit expression (14) is obtained for the number of p -combinations satisfying the conditions k≤d j ≤k′ , j =1(1) p −1, l≤n−d≤l′ . Recurrence relations are found. Special cases include generalized Fibonacci and Lucas numbers.