December 2005 - mathfun@mailman.xmission.com

[math-fun] Dick Howell
by Schroeppel, Richard 04 Dec '05

04 Dec '05

(Apologies to the non-MIT folks here, but for historical reasons we have a lot of MIT PDP1 & AILab people.) The MIT Class Notes (nominally part of Tech Review, but nowadays it's email) reports that Richard Howell '68 died in August. Class Notes says they have no other information. Dick Howell was part of the PDP1 crowd ~1965-70. I remember he wrote a Conway Life program for the PDP1, and an early chess program. Rich rcs(a)cs.arizona.edu

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[math-fun] A few self-descriptive fractal sequences
by Kerry Mitchell 04 Dec '05

04 Dec '05

"Lower trimming" is one technique for determining if a sequence is a fractal sequence. 1 is subtracted from each element, and all the zeroes are thrown away. If the resulting sequence is the same is the original, then it is a fractal sequence. This is an example of a fractal sequence: 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, ... If we subtract 1 from each element, we get: 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, ... and removing all the zeroes leaves us with: 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, ... Which is the beginning of the original sequence. This sequence is also self-descriptive, in that each element gives the number of zeroes that were removed before it. And, for reasons I haven't figured out, the indices where the sequence hits a new maximum value (2 at the 2nd position, 3 at the 5th position, 4 at the 13th, 5 at the 34th, etc.) are every second Fibonacci number. Here's another one: 1, 2, 1, 3, 2, 1, 4, 1, 3, 2, 5, 1, 2, 4, 1, 3, 1, 6, 2, 3, 5, 1, 2, 1, 4, 2, 7, 1, 3, 4, 1, 6, ... This sequence gives the number of numbers that are retained between zeroes that are dropped. Alternatively, each element is the number of numbers between 1's in the original sequence. This last fractal sequence combines both ideas. 1, 2, 1, 1, 3, 2, 1, 2, 1, 4, 1, 1, 1, 3, 2, 3, 1, 1, 2, 5, 1, 2, 1, 1, 2, 2, 1, 4, 1, 1, 1, 1, ... Each element gives the number of zeroes dropped before it and the number of elements written before the next zero. Nothing too terribly earth-shaking, but enough to keep me occupied on a Saturday night. :-) Kerry

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[math-fun] Periodic Antisymmetric Matrices counted by Hermitte Numbers
by wouter meeussen 03 Dec '05

03 Dec '05

Antisymmetric: Transpose(A)= -A, so zero diagonals Periodic: some matrix power = Identity Matrix A067994= Hermitte Numbers= 1,0,2,0,12,0,120,0,1680, ... Their periods are all 4. Strong law of small numbers checked upto n=8. Who says this is self-evident? W.

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[math-fun] Algorithm for counting nonisomorphic colorings
by David Wilson 03 Dec '05

03 Dec '05

Tsvetelina Petkova has asked for my help with a program to count nonisomorphic colorings of general graphs. I haven't made any promises. I can describe an algorithm for doing this, however, it would be a bear to program in Perl or C++, which are all I have at my disposal. I was wondering if someone versed in Mma or some other symbolic language might want to discuss the possibility of implementing my algorithm. -------------------------------- - David Wilson

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RE: [math-fun] New sequences; comments on old; origin of problem?
by dasimov＠earthlink.net 02 Dec '05

02 Dec '05

Richard Guy writes many cheerful facts about the square of n by n dots: << Given a square n by n array of dots, what is the maximum number of edges in a convex polygon whose vertices are members of the array? >> As usual, I wonder about the same question applied, instead of to the n by n array of dots, to the n by n "discrete square torus" array of dots. I.e., if all addition is mod n, then each point (K,L) of Z/nZ x Z/nZ is connectable to (K+d, L+e) where (d,e) lies in the set {-1,0,1} x {-1,0,1} - {0,0}. Another related question is to use the "discrete rhombic torus", also describable as the points Z/nZ x Z/nZ, but with the point (K,L) connectable to (K+d,L+e) only for (d,e) in {-1,0,1}x{-1,0,1} - {(-1,-1),(0,0),(1,1)} (again, addition is mod n). --Dan

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