we first show that such a sequence cant have more than 17 or more numbers

let us assume that the numbers are a1,a2,a3-----a17

    a1 a2 a3 a4 a5 a6 a7

    a2 a3 a4 a5 a6 a7 a8

    a3 a4 a5 a6 a7 a8 a9

     ----------------------------

    a11 a12 a13 a14 a15 a16 a17

the sum of the terms in  positive  in every column and negative in every row

if by adding all rows it is positive and adding all columns it is negative

which is a contradiction

so we can have at most 16

 terms

which is poosible by 8,8,-21,8,8,8,-21,8,8,-21,8,8,8,-21,8,8