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.ur...@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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> Edu-sig@python.org
> http://mail.python.org/mailman/listinfo/edu-sig
>
>
--
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/
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