Clark's Triangle


Clark's triangle is a number triangle created by setting the vertex equal to 0, filling one diagonal with 1s, the other diagonal with multiples of an integer f, and filling in the remaining entries by summing the elements on either side from one row above. The illustration above shows Clark's triangle for f=6 (OEIS A090850).

Call the first column n=0 and the last column m=n so that


then use the recurrence relation


to compute the rest of the entries. The result is given analytically by

 c_(mn)=f(m; n+1)+(m-1; n-1),

where (n; k) is a binomial coefficient (M. Alekseyev, pers. comm., Aug. 10, 2005).

The interesting part is that if f=6 is chosen as the integer, then c_(m2) and c_(m3) simplify to


which are consecutive cubes (m-1)^3 and nonconsecutive squares n^2=[(m-1)(m-2)/2]^2.

The sum of the mth row for m>0 is given by


(M. Alekseyev, pers. comm., Aug. 10, 2005).

Binary plot for Clark's triangle

The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Clark's triangle with f=6.

See also

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

Explore with Wolfram|Alpha


Clark, J. E. "Clark's Triangle." Math. Student 26, No. 2, p. 4, Nov. 1978.Sloane, N. J. A. Sequence A090850 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Clark's Triangle

Cite this as:

Weisstein, Eric W. "Clark's Triangle." From MathWorld--A Wolfram Web Resource.

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