The Rational Theorist: The Dot Pyramid Sequence and The Sum of the Diagonal Dot Pyramid Spiral

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Friday, February 18, 2011

The Dot Pyramid Sequence and The Sum of the Diagonal Dot Pyramid Spiral

Just wanted to crack this little mystery is all. The dot pyramid sequence relates to finding the combinations of "k" points that can occupy a given number of "n" spots assuming any amount of points can occupy the same spots, ergo it relates to possible ground state patterns that bosons can assume in Einstein-Bose Condensation at cool temperatures. However, this concept is also of pure aesthetic mathematical values as well.

Note - Click Images to Enlarge

What is that mysterious diagonal sum sequence though? Here is cracking of that mystery, apparently the points that make up each sum-sequence take on a family of parabolic curves of the form "Rx(i)= -2i^2 + (x+3)i - x" where the number of positive integer values for each curve goes from "i=1" to "i=(x+(1+(-1)^(x+1))/2)/2", and each sequential diagonal sum is the summation for Rx(i) over the range of "i" for each Matrix row "Rx" pertaining to each parabola within the family of parabolas.

The thing that caught my eye about that sum of diagonals of dot pyramid sequence (SDDP) is that it looks similar to the Fibonacci sequence (1,1,2,3,5,8,13,21...) except that it diverges short of that sequence after 8... "SDDP = (1,1,2,3,5,8,12,18,25,35,46,...)", so just for curiosity sake I decided to graph this lesser known spiral out on Excell:

Note - Click Image to Enlarge