Clark,s Triangle
المؤلف:
Sloane, N. J. A.
المصدر:
Sequence A090850 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
7-1-2021
1351
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 
, 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 
(OEIS A090850).
Call the first column 
and the last column 
so that
then use the recurrence relation
 |
(3)
|
to compute the rest of the entries. The result is given analytically by
 |
(4)
|
where 
is a binomial coefficient (M. Alekseyev, pers. comm., Aug. 10, 2005).
The interesting part is that if 
is chosen as the integer, then 
and 
simplify to
which are consecutive cubes 
and nonconsecutive squares ![n^2=[(m-1)(m-2)/2]^2](https://mathworld.wolfram.com/images/equations/ClarksTriangle/Inline22.gif)
.
The sum of the 
th row for 
is given by
 |
(7)
|
(M. Alekseyev, pers. comm., Aug. 10, 2005).
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 
.
REFERENCES:
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."
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
