History[ edit ] Chess composer Max Bezzel published the eight queens puzzle in Franz Nauck published the first solutions in
What do you get when you methodically build a Lisp on top of symbolic replacement semantics? You get the Mathematica language, of which Mathematica and Mathics appear to be the only incarnations.
Let's say you forgot how to multiply matrices. Well, just type in some symbols and see the results empirically: In fact if you have a Tron zapper you can even zap your cat into Mathematica and have him fill up one of those matrix slots, for the advancement of science.
Our neighbor will miss him. The exponential identity for the Pascal matrix is not difficult to understand based on the series definition of the exponential function: You can see that the diagonal gets multiplied by subsequently shifted versions of itself, so the calculation ends up creating factorial products.
The binomial coefficient itself is of course directly related to Pascal's triangle. Also notice that every power of the matrix has its numbers on a different diagonal, so when we sum up all the powers there is no interaction to account for. Every term in the series is a distinct diagonal of Pascal's triangle.
I didn't find anything interesting along this line though.
What about graphs represented by the Sierpinski matrix itself? It has some pretty symmetries. I did some tiresome work trying to figure out what polyhedron it might be.
It's the tetrakis hexahedron: And look, we can run this polyhedron grapherizer willy-nilly allabouts, like on the Archimedean solids: For one, the powers of the Sierpinski matrix are Sierpinski matrices!
This isn't necessarily interesting though. The powers of a triangular matrix are going to be triangular. But the numbers follow a curious sequence of powers. And this sequence occurs in every column and every row of the matrix, if you hop over the zeros.
We can normalize the powers to find: This power sequence appears in OEIS as the number of ones in the binary representation of namong other descriptions.
Here is a totally practical application of all of this. A pretty array of buttons: Thus the game is ultimately a quaint philosophical remark on the roles of the sexes.After compilation of the script (on the PC), it can be tested with the leslutinsduphoenix.com debugger supports breakpoints, single-step, a disassember, the trace history, and a display for the symbol table with variable values.
To simplify the development of scripts, the editor contains several tools explained in . By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is .
Write a program leslutinsduphoenix.com that reads in two files specified at the command line one line at a time, computes the LCS on the sequence of constituent lines of each file, and prints out any lines corresponding to non-matches in the LCS.
a) Write an explicit formula for finding the nth term of the sequence. b) thUse this formula, to find the 8 term of the sequence. 10) Write an explicit formula of the geometric sequence that has two given terms. We would like to find a formula for f(n) in terms of leslutinsduphoenix.com this case the pattern is fairly easy to see: each value of f(n) is 3 more than the previous leslutinsduphoenix.com can see this if we look at the differences between successive numbers in the sequence, writing these differences in a row beneath the f(n) row.
"Ah, that makes sense." You say. Indeed, but what's cool is that we then have a pedantic way of specifying the Sierpinski triangle.