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Ramsey 2879

  1. Jan 6, 2007 #1
    The Wythoff Array (http://www.research.att.com/~njas/sequences/A035513) is a square array of Fibonacci like sequences T(i,j) = T(i,j-1) + T(i,j-2) where T(i,1) = i and T(i,2) = Floor(i*(sqrt(5)+1)/2). Each row of the Wythoff Array has a characteristic number A(i) which is a constant that is equal to value of T(i,2n)^2 - T(i,2n-1)*T(i,2n+1). If the index n is off by one, i.e. even-odd-even instead of odd-even-odd we get A(i) = T(i,2n)*T(i,2n+2) - T(i,2n+1)^2
    The sequence A(i) appears to be identical to that at http://www.research.att.com/~njas/sequences/A022344 although
    I dont have a proof.
    No more than two primitive rows (every two adjacent terms are coprime) share the same characteristic value and the second row is just the extension of the sequence in the first row T(i,j) beginning with the pair [T(i,2),-T(i,1)] which is extended until the two terms [x, Floor(x*(sqrt{5}+1)/2)] are reached.

    Now Horadam has previously published a result concerning Fibonnacci series and Pythagorean triples "Fibonacci Number Triples" Amer. Math. Monthly 68(1961) 751-753. That paper shows that if F(0), F(1), F(2) and F(3) are 4 sequential numbers of a Fibonacci type sequence then P = (2F(1)*F(2),F(0)*F(3),2F(1)F(2)+F(0)^2) is a Pythagorean triplet. That is (2F(1)*F(2))^2 + (F(0)*F(3))^2 = (2F(1)*F(2)+F(0)^2)^2. Primative Pythagorean triplets are generated from primative Fibonacci sequences.

    I checked as to what sequence I would get if H(i) = T(i,j)^2 + 2(T(i,j+1)*T(i,j+2). In each case the sequence is a bijection of a primitive Fibonnacci sequence F(i) multiplied by G where G is the greatest common factor of any two adjacent terms, that is H(i) =G*F(2i-1). By filling in the intermediate values F(2i) = (H(i+1)-H(i))/G it is possible to compute the characteristic values C(i) of these new Fibonacci sequences. Surprisingly, I got the following sequence:

    1,1,1,1,1,121,361,121,1,961,361,961,1,841,121,1,... These terms are all 1 or the square of a prime or of a product of primes ending in 1 or 9. The primes occur at the same positions as a factor of A(i) in the characteristic sequence that I mentioned above as in the new sequence C(i). Do you find this to be a coincidence or a remarkable discovery?
    Last edited: Jan 7, 2007
  2. jcsd
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