Conference rusure::math

    After looking at the first few hundred lines of this triangle, I
    have another conjecture:
    	The 2^n+1st line of this triangle contains three odd numbers
    	(the 1's at each end and the number in the middle); the rest
    	are even.

    re. .1
    
    can you include these few hundres lines in here, or maybe the algorithm
    which generetes the numeric triangle, or even MAPPLE's function?
    
    kgg
    

The following BASIC program displays the parity of the right half of the 
    triangle for the first 1000 lines. (Hmmm. I can't get rid of the
    spaces in front of the first line below. Please delete them yourself.)
    
    10	declare byte a(1000), b(1000)
	declare integer i, j
20	a(1) = 1
	b(1) = 1
	print  a(1)
30	for i = 2 to 1000
		a(i) = 0
	next i
40	for i = 2 to 1000
		for j = 2 to i
			b(j) = a(j-1) XOR a(j) XOR a(j+1)
		next j
		for j = 2 to i
			a(j) = b(j)
		next j
		print i
		for j = 1 to i
			if (j/25%)*25% = j
			then print a(j) 
			else print a(j); 
			end if
		next j
		print
		print
	next i

    
    	What is MAPLE?
    
    	David
    

re. .4

This is from the DCL  $ help maple

MAPLE

   Maple is an interactive program which can algebraically manipulate
   unbounded integers, exact rational numbers, real numbers with arbitrary
   precision, symbolic formulae, polynomials, sets, lists, and equations.
   It can solve systems of equations, differentiate formulae, and
   integrate formulae.  For information on Maple, see "Maple: A Sample
   Interactive Session" or "First Leaves: A Tutorial Introduction  to
   Maple" or "MAPLE Reference  Manual".

   Format:

   	MAPLE [-s] [-l] [-b directory-name] [-q] [< File_Spec] [> File_Spec]

  Additional information available:

  Commands   Library    Parameters Initialization        Files      Interrupts VMS_commands


Enjoy,

Kostas G.

re 554.1				  n
					 2
    This is equivalent to proving that (   ) is odd only when
    					 k
	    n      n-1
    k = 0, 2 , or 2   .  See note 558.4 for a hint on proving this.

    Well,
    
      lets see if can contribute to the solution of the problem in .0.
    
    The problem was: In the numerical triangle prove that in every row
    beginning with the third, there is an even number.
    
                             1
                          1  1  1
                       1  2  3  2  1          3rd row
                    1  3  6  7  6  3  1         4th row
                 1  4 10 16 19 16 10  4  1        5th row
              1  5 15 30 45 51 45 30 15  5  1       6th row

    
    Denote the even numbers by the letter  E  and the odd numbers by
    the letter  O.  The first 4 elements of the 3rd row have the notation
    OEOE, the fourth  OOEO,  the fifth  OEEE,  the sixth  OOOE, and
    the seventh  OEOE. 
    
            
    
                             1
                          1  1  1
                       O  E  O  E  1          3rd row
                    O  O  E  O  6  3  1         4th row
                 O  E  E  E 19 16 10  4  1        5th row
              O  O  O  E 45 51 45 30 15  5  1       6th row
    
    
    The cycle then repeats, ( the first  4  elements of each row are 
    determined by the first four elements of the preceding row). 
    There will therefore be at least one even number in every row.
    
    
    Enjoy,
    
    Kostas G.
    
    

    You get some interesting patterns by examining the residues modulo
    3 of the numbers in this triangle. Hypothesis: the numbers in the
    sum(3**n)-th row are all congruent to 1 modulo 3.