## Clark's Triangle

A Number Triangle created by setting the Vertex equal to 0, filling one diagonal with 1s, the other diagonal with multiples of an Integer , and filling in the remaining entries by summing the elements on either side from one row above. Call the first column and the last column so that

 (1) (2)

then use the Recurrence Relation
 (3)

to compute the rest of the entries. For , we have
 (4)

 (5)

For arbitrary , the value can be computed by Summing this Recurrence,
 (6)

Now, for we have
 (7)

 (8)

so Summing the Recurrence gives

 (9)

Similarly, for we have

 (10)

Taking the Sum,
 (11)

Evaluating the Sum gives
 (12)

So far, this has just been relatively boring Algebra. But the amazing part is that if is chosen as the Integer, then and simplify to

 (13) (14)

which are consecutive Cubes and nonconsecutive Squares .

See also Bell Triangle, Catalan's Triangle, Euler's Triangle, Leibniz Harmonic Triangle, Number Triangle, Pascal's Triangle, Seidel-Entringer-Arnold Triangle, Sum

References

Clark, J. E. Clark's Triangle.'' Math. Student 26, No. 2, p. 4, Nov. 1978.