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HopD
Posts:
14
Registered:
12/6/04


Shapes of binomial #s
Posted:
Jan 19, 1995 8:24 PM


This is a conjecture about the shapes of numbers that are coefficients of the different terms in (a+b)^n
Below I've attempted to draw the first four coefficients of (a+b)^6 (which are 1, 6, 15, and 20)
1 1 1 1 1 1 1 1 11 11 11 11 1 111 111 111 1 1111 1111 1 11111 1
As is well known, the third coefficient is a triangle number. Looking at these numbers I believe the first coeeficient could be described as a point number and the second as a line number.
The fourth coefficient is the sum of consecutive triangle numbers. If these triangles are stacked on top of each other they form a tetrahedron. (I have a more clear rendition of this in the JPEG format. If you would like to see it, let me know via email)
To me this suggests a sequence:
The first coefficient of a binomial expansion is a point number It is one point.
The second coefficient is a line number Lines are bound by 2 points
The third is a triangle number It is bound by 3 lines.
The fourth, a tetrahedron number. Tetrahedrons are bound by 4 triangles.
To me this suggests a sequence. Would the next coefficient be a number that forms an object in 4space? Presumably this object would be bound by 5 tetrahedrons. The number after that would be a 5 space object bound by 6 whachamacallits, etc. etc.
Can anyone prove or disprove this conjecture?



