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Ahmad Kheder Major: MATHEMATICS Advised: Dr STEFAN FORCEY

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we choose starting terms a1 … ak and then determine each later term

a1 always equals zero

a2 is any number, or zero

a3 you pick any number that is greater than or equal to a1 + a2 = a2

For a2 to ak, must obey the rule that an greater than or equal to max {ai + an-i} for i=1…n-1

Then for n > k we define an = max {ai + an-i} for i=1…n-1

Is there a shortcut?

Description of the problemFor example if the starting points are 0, 1, 2, 4, and 9 then what are the next five terms?

exampleThis is the Mathematica program to get the 1st 25 terms

clear[a]a[1]=0;a[2]=1;a[3]=2;a[4]=4;a[5]=9;a[n_]:=a[n]=Max[Table[[email protected][email protected](n-i),{i,1,n-1}]];Table[[email protected],{n,1,25}]

Mathematica codeIf the starting points are 0, 1,2, 4, and 9 then the next terms can be calculated by adding the last given term to the first to give the a6 term then add the last term to second term to get a7 and add last term to third term to get a8 and add last term to fourth term to geta9 and add last term to itself to get a10

We found 0,1,2,4,9 by evaluating f(x) at the integers, for f(x) continuous, differentiable, increasing, and concave up on (1, 5).

Is there a shortcut?By concave up or flat we mean f’’(x) ≥ 0

for any two ai , aj where 1 ≤ i < j ≤ k

ah for i ≤ h ≤ j lies below or on the line connecting aiand aj

The shortcut says that an= pak + aq where n=pk+q, 0 ≤q<k

Starting GraphWe conjecture that to have the shortcut the starting points should be found by evaluating f at the integers,

for f continuous, differentiable, increasing, and concave up on (1, k).

If for ai , aj where 1 ≤ i < j ≤ k, we have that ah for

i ≤ h ≤ j lies on or below the line connecting aiand aj

Then an= pak + aq where n=pk+q, 0 ≤q<k

Conjecture / TheoremOur sequences are examples of Operads.

Illustrate the Vincent Ferreiro, Jack F. Douglas, James Warren, and Alamgir Karim

Measurements they took in 2002 as certain crystals formed in solution. Their first

study was

Crystals gROWTHGraph of experimental Crystal growth is very similar to one of our graphs

Graph of sinusoidal model of Crystal growth is very similar to one of our zigzag series of graphs C0,0,0,0,0,1,2,3,4,5,6

Dr. STEFAN FORCEY’s N-fold operads research

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