DATA "0426597138","2365478109","2581470369","8103627495"
CLS
FOR seq = 1 TO 4
  PRINT seq
  READ s$
  FOR i = 1 TO 10
    dig(i) = VAL(MID$(s$, i, 1))
  NEXT
  FOR a = 1 TO 9
   FOR b = 0 TO 9
    good = 1
    FOR i = 1 TO 9
      v = (a * dig(i) + b) MOD 10
      IF v <> dig(i + 1) THEN good = 0
    NEXT
    IF good THEN PRINT a, b
   NEXT
  NEXT
  PRINT : PRINT
NEXT seq

finds that only the third column has a linear relation mod 10, and is in fact the sequence given in the example, when using the up arrow.

The others must not be of the form f(x)= (a*f(x-1)+b) mod 10.