cp's OEIS Frontend

From Kenneth J Ramsey, 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 squarefree portion of a(n). All other factors of a(n) appear only as squares.

"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.

"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.

"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)

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

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

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

Every Fibonacci sequence with positive terms occurs as some row of Wythoff's array (A035513), so a(n) is always defined. There appear to be two possible offsets for the sequence of sums within row a(n); either T(n,j-1) + T(n,j+1) = T(a(n),j-3) for j>=3 or T(n,j-1) + T(n,j+1) = T(a(n),j-1) for j>=1. The first case seems to occur whenever n has a Zeckendorf representation which ends in 1, 10100, 101000100, 1010001000100, 10100010001000100, etc. (each successive ending is obtained by changing the left-hand 1 to 10100). The values of these endings are 1,11,79,545,3739,25631,175681 ... and equal F(i)*F(i+1) + F(i+2)^2 for i = 0,2,4,6,... where F(i) is the i-th Fibonacci number. These values also appear in the table A127561 at a(1,0), a(1,1), a(2,3), a(5,8), ..., a(F(2n-1),F(2n)) for n = 0,1,2,3....

The Zeckendorf representation of n is the unique binary sequence ...,b(4),b(3),b(2) for which n = sum_{i>=2} b(i)F(i) and two consecutive b's cannot both be 1. For example, the Zeckendorf representation of 100 is 1000010100, since 100 = 89+8+3 = F(11)+F(6)+F(4).