Quotient-Difference Table
A quotient-difference table is a triangular Array of numbers constructed by drawing a sequence of 
numbers in a horizontalrow and placing a 1 above each. An additional ``1'' is then placed at the beginning and end of the row of 1s, and the value ofrows underneath the original row is then determined by looking at groups of adjacent numbers
and computing
for the elements falling within a triangle formed by the diagonals extended from the first and last ``1,'' as illustrated above.
0s in quotient-difference tables form square ``windows'' which are bordered by Geometric Progressions. Quotient-difference tables eventually yield a row of 0s Iff the starting sequence is defined by a linearRecurrence Relation. For example, continuing the above example generated by the Fibonacci Numbers
and it can be seen that a row of 0s emerges (and furthermore that an attempt to extend the table will result in division by zero). This verifies that the Fibonacci Numbers satisfy a linear recurrence, which is in fact given by thewell-known formula
However, construction of a quotient-difference table for the Catalan Numbers, MotzkinNumbers, etc., does not lead to a row of zeros, suggesting that these numbers cannot be generated using a linearrecurrence.See also Difference Table, Finite Difference
References
Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 85-89, 1996.
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