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Topic: Problem involving sequence(CON)
Replies: 0

 Natthapol Wongsaroj Posts: 3 Registered: 12/12/04
Problem involving sequence(CON)
Posted: Jun 17, 1996 7:15 AM

Last time I 've post a problem as follow :
Given two positives integers p and q , Generates the longest
possible sequence of integer numbers which sum of any p consecutive
is positive and sum of any q consecutive is negative.

I 've got the solutions but I 'm not sure if it 's right or not.
Can anyone verify the following method?

First : The length of the sequence is less than p + q - gcd(p,q) for sure
(Anyone who want to know how to prove this please mail me)

To simplify the problem add another variable n ( = length of the
sequence) for the given numbers n,p and q ,this problem can be modeled as
a system of equations. for example if (n,p,q) = (5,3,4) the following
equations will solve the problem.
Let the sequence be x[1],x[2],x[3],...,x[5]
The equations are
x[1]+x[2]+x[3] = c1
x[2]+x[3]+x[4] = c2
x[3]+x[4]+x[5] = c3
x[1]+x[2]+x[3]+x[4] = -c4
x[2]+x[3]+x[4]+x[5] = -c5

and c1,c2,...,c5 are positive integers
I 've tried to solve this problem by programming. I 've found it
works for arbitary value of c1,c2,c3,c4 and c5. And as far as I 've tested
my program I 've found that the longest possible length is always =
p + q - gcd(p,q) - 1. But anyway I couldn't prove that arbitary positive
integers can be used for ci and the longest possible length is as I 's
mentioned. Can anyone verify it? Or is there a better way to solve this
problem?