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A022344 Allan Wechsler's "J determinant" sequence.

%I A022344 #41 Aug 20 2025 10:34:16
%S A022344 1,5,4,9,16,11,19,11,20,31,19,31,45,29,44,25,41,59,36,55,29,49,71,41,
%T A022344 64,89,55,81,44,71,100,59,89,121,76,109,61,95,131,79,116,61,99,139,80,
%U A022344 121,164,101,145,79,124,171,101,149,76,125,176,99,151,205
%N A022344 Allan Wechsler's "J determinant" sequence.
%C A022344 From _Kenneth J Ramsey_, Jan 06 2007: (Start)
%C A022344 "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 squarefree portion of a(n). All other factors of a(n) appear only as squares.
%C A022344 "Moreover, the squarefree 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 squarefree portion of a(n) except without the factor of 5.
%C A022344 "For example the Lucas 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 Lucas 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 J Ramsey_, Feb 11 2007
%C A022344 A product p of rows n and m of the Wythoff Array, such that a(p) = a(n)*a(m) as described in the conjecture above, is defined by A357097(n, m). - _Peter Munn_, Aug 15 2025
%C A022344 a(n) = |T(n,i)^2 - T(n,i-2)*T(n,i+2)| for all i > 2, where T = Wythoff array.  Indeed, if k > 0, then |T(n,i)^2 - T(n,j-k)*T(n,j+k)| = (F(k)^2)*a(n) for j > k.  That is, if m is in this sequence, then 4*m, 9*m, 25*m, 64*m, ... are also in this sequence.  - _Clark Kimberling_, Jul 15 2016
%D A022344 _Allan C. Wechsler_, posting to math-fun mailing list Dec 04 1996.
%H A022344 Michael De Vlieger, <a href="/A022344/b022344.txt">Table of n, a(n) for n = 0..10000</a>
%H A022344 A. F. Horadam, <a href="http://www.jstor.org/stable/2311978">Fibonacci Number Triples</a>, The American Mathematical Monthly, Vol. 68, No. 8 (Oct., 1961), pp. 751-753.
%F A022344 a(n) = floor((n+1)*tau)^2 - n*floor((n+1)*tau) - n^2.
%F A022344 a(A357097(x,y)) = a(x) * a(y). - _Peter Munn_, Aug 15 2025
%p A022344 Digits := 50: t := evalf((1+sqrt(5))/2): f := n->floor( n*t)^2-(n-1)*floor(n*t)-(n-1)^2:
%t A022344 Table[#^2 - n # - n^2 &[Floor[(n + 1) GoldenRatio]], {n, 0, 51}] (* _Michael De Vlieger_, Jun 30 2016 *)
%o A022344 (Magma) [Floor((n+1)*((1+Sqrt(5))/2))^2-n*Floor((n+1)*(1+Sqrt(5))/2)-n^2: n in [0..60]]; // _Vincenzo Librandi_, Jul 01 2016
%Y A022344 Cf. A035513, A127561, A275068, A357097.
%Y A022344 Range of values: A031363.
%K A022344 nonn
%O A022344 0,2
%A A022344 _N. J. A. Sloane_