""" Mosaics using Python generators... by K. Urner 4dsolutions.net based on a thread on mathfuture: http://groups.google.com/group/mathfuture/msg/9d098696c2ea7426?hl=en Just as the Fibonacci sequence generator might be "seeded" by any two integers to start off, so might Pascal's Triangle be seeded with any beginning row. When these sequences are formed into tabular or other areal displays, such that information may be gleaned from rows and columns, these formats may be called "mosaics" in some of the literature. """ def fibonacci( a=0, b=1): """ Generator for a Fibonacci sequence starting from any two integers See: Online Encyclopedia of Integer Sequences http://oeis.org/A000045 >>> gen = fibonacci() >>> [next(gen) for i in range(11)] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55] >>> gen = fibonacci(-1, 2) >>> [next(gen) for i in range(11)] [-1, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76] """ while True: yield a b, a = a + b, b def pascal( therow = [1]): """ Generator for Pascal's Triangle starting from any top row Enter list for top row See: Number Mosaics by Adi R. Kanga (pg. 29, Fig 20) http://bit.ly/gDmmAo >>> gen = pascal([15, 20, 6]) >>> next(gen) [15, 20, 6] >>> next(gen) [15, 35, 26, 6] >>> next(gen) [15, 50, 61, 32, 6] >>> next(gen) [15, 65, 111, 93, 38, 6] >>> next(gen) [15, 80, 176, 204, 131, 44, 6] >>> next(gen) [15, 95, 256, 380, 335, 175, 50, 6] """ while True: yield therow therow = [ i + j for i, j in zip(therow + [0], [0] + therow) ] def _test(): import doctest doctest.testmod() if __name__ == "__main__": _test()
Your generalized Fibonacci sequences and generalized Pascal's Triangles are linear combinations of the standard sequences and triangles with shifts. m,n,m+n,m+2n, 2m+3n... is the sum of the sequences m,0,m,m,2m,... 0,n,n,2n,3n,... and similarly, the coefficients of x, y, and z in this triangle are all Pascal numbers. x, y, z x, x+y, y+z, z x, 2x+y, x+2y+z, y+2z, z x, 3x+y, 3x+3y+z, x+3y+3z, y+3z, z On Fri, Feb 4, 2011 at 15:10, kirby urner <kirby.urner@gmail.com> wrote:
""" Mosaics using Python generators... by K. Urner 4dsolutions.net based on a thread on mathfuture: http://groups.google.com/group/mathfuture/msg/9d098696c2ea7426?hl=en Just as the Fibonacci sequence generator might be "seeded" by any two integers to start off, so might Pascal's Triangle be seeded with any beginning row. When these sequences are formed into tabular or other areal displays, such that information may be gleaned from rows and columns, these formats may be called "mosaics" in some of the literature. """ def fibonacci( a=0, b=1): """ Generator for a Fibonacci sequence starting from any two integers See: Online Encyclopedia of Integer Sequences http://oeis.org/A000045
>>> gen = fibonacci() >>> [next(gen) for i in range(11)] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55] >>> gen = fibonacci(-1, 2) >>> [next(gen) for i in range(11)] [-1, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76] """ while True: yield a b, a = a + b, b
def pascal( therow = [1]): """ Generator for Pascal's Triangle starting from any top row Enter list for top row See: Number Mosaics by Adi R. Kanga (pg. 29, Fig 20) http://bit.ly/gDmmAo
>>> gen = pascal([15, 20, 6]) >>> next(gen) [15, 20, 6] >>> next(gen) [15, 35, 26, 6] >>> next(gen) [15, 50, 61, 32, 6] >>> next(gen) [15, 65, 111, 93, 38, 6] >>> next(gen) [15, 80, 176, 204, 131, 44, 6] >>> next(gen) [15, 95, 256, 380, 335, 175, 50, 6] """ while True: yield therow therow = [ i + j for i, j in zip(therow + [0], [0] + therow) ]
def _test(): import doctest doctest.testmod() if __name__ == "__main__": _test()
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-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://www.earthtreasury.org/
participants (2)
-
Edward Cherlin -
kirby urner