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February 11th, 2017, 09:48 PM  #1 
Newbie Joined: Feb 2017 From: Bay Area, US Posts: 5 Thanks: 0 Math Focus: 3x+1  Summing a set past the end of the set.
The sum from n=1 to k+d of a_n is equal to a_k. We can deduce that a_n = a_(nd)  a_(nd1) and if a_1 = 1 and d = 1 it results in: [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,...] When we change the value of d, it becomes more chaotic. When d=4, we get: [1,0,0,0,0,1,0,0,0,1,1,0,0,1,2,1,0,1,3,3,1,1,4,6,4,2,5,10,10,6,7,...] Some of you who are keen may notice Pascal's triangle hidden in there, and in fact, when d approaches infinity, it will give us alternating numbers in pascal's triangle, separated by infinite zeros. But these are trivial. The sum from n=1 to dk of b_n is equal to b_k. We set d=2 the first 4 are [1,0,x,y] where x+y=1, so there are infinitely many routes we could go. The simplest being x=1, y=0 so, the first 6 are [1,0,1,0,x',y'] where x'+y'=1, using the same logic, the first 8 are [1,0,1,0,1,0,x'',y''] where x''+y''=0, using x''=y''=0 ... using the simplest solution we get [1,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,...] Most likely never getting a digit above 1, due to y*=0 and b_n  b_(n+1)<=1 When we add the rule that x*=y* it might be different but I'm much too tired to pay any attention. As a final note, I apologize if this is already a thing and I'm taking credit for it, if it is, I'd like to learn more. Last edited by skipjack; February 12th, 2017 at 01:54 AM. 

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end, past, set, summing 
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