“Clark's Triangle”的概念、定义、习题解题方法及翻译-数学辞典

单词

Clark's Triangle

释义

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 oneither 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.

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