%I A022344
%S A022344 1,5,4,9,16,11,19,11,20,31,19,31,45,29,44,25,41,59,36,55,
%T A022344 29,49,71,41,64,89,55,81,44,71,100,59,89,121,76,109,61,
%U A022344 95,131,79,116,61,99,139,80,121,164,101,145,79,124,171
%N A022344 Allan Wechsler's "J determinant" sequence.
%C A022344 Comments from Kenneth Ramsey (RamseyKK2(AT)aol.com), Jan 06 2007 (Start) 
               "a(n) = the characteristic value of the row T(n,i) of the Wythoff 
               array A035513 which is the absolute value of T(n,i)^2 - T(n,i-1)*T(n,
               i+1). Only the number 5 or prime factors ending in 1 or 9 form the 
               square-free portion of a(n). All other factors of a(n) appear only 
               as squares.
%C A022344 "Moreover, the square-free portion (less the factor 5) squared is the 
               characteristic value of the Fibonacci sequence whose bijection relates 
               to c term of the Horadam "Fibonacci Number Triples" Amer. Math. Monthly 
               68(1961) 751-753. That paper showed that if F(0), F(1), F(2), F(3) 
               are 4 sequential numbers in a row of the Wythoff array, then P = 
               (2F(1)*F(2),F(0)*F(1),2F(1)*F(2) + F(0)^2) is a Pythagorean triple 
               (a,b,c) i.e. a^2 + b^2 = c^2.
%C A022344 "If i varies and c(n,2i-1) = F(n,i)^2 + 2F(n,i+1)*F(n,i+2) and C(n,2i) 
               is set to equal C(n,2i+1)-C(n,2i-1) then, the sequence F(x,i) = C(n,
               i)/G, where G is the greatest common divisor of the adjacent terms 
               C(n,i), is a Fibonacci sequence having the characteristic value which 
               is the square of the square-free portion of a(n) except without the 
               factor of 5.
%C A022344 "For example the Lucus sequence or the second row of the Wythoff array 
               has the characteristic value of A(2) = 5 and the C(n,i) terms are 
               each 5 times the sequential terms 34,89,233,... which is a bijection 
               of the terms in the 1st row of the Wythoff array which row has the 
               characteristic value of 1. This is so even though adjacent terms 
               of the Lucus sequence are coprime." (End)
%C A022344 Conjecture: Every pair of Fibonacci sequences, F1 and F2, appear in rows 
               n and m of Wythoff's Array, respectively and have respective characteristics 
               a(n) and a(m). Also, there is a third Fibonacci sequence F3, defined 
               by F3(i) = F1(i) * F2(j+1) - F1(i+1)*F2(j) where j is held constant. 
               The sequence F3 appears in row p of Wythoff's array and has the characteristic 
               a(p) = a(n)*a(m). - Kenneth Ramsey (RamseyKK2(AT)aol.com), Feb 11 
               2007
%D A022344 Allan Wechsler (acw(AT)alum.mit.edu), posting to math-fun mailing list 
               Dec 04 1996.
%F A022344 [ (n+1)*tau ]^2 - n*[ (n+1)*tau ] - n^2.
%p A022344 Digits := 50: t := evalf((1+sqrt(5))/2): f := n->floor( n*t)^2-(n-1)*floor(n*t)-(n-1)^2:
%Y A022344 Cf. A035513.
%Y A022344 Cf. A127561.
%Y A022344 Sequence in context: A110617 A102081 A068397 this_sequence A046588 A086654 
               A152064
%Y A022344 Adjacent sequences: A022341 A022342 A022343 this_sequence A022345 A022346 
               A022347
%K A022344 nonn
%O A022344 0,2
%A A022344 N. J. A. Sloane (njas(AT)research.att.com).