Permalink Submitted by Anonymous on February 4, 2016

Hi Rachel. This is to continue our interesting discussion about a result with squares I'd thought I'd found in the Fibonacci sequence (see below). This time compare some of the sequence to the left as well as the right of 0 with a corresponding segment of the famous Lucas sequence:

The Lucas can be seen as resulting from swapping round two consecutive Fibonacci terms, from 2, -1 to -1, 2 while retaining the same addition rule as Fibonacci, adding two consecutive numbers to get the third as you go right. To the right in the Lucas we now have not just one but two integer squares, those of -1 and 2, namely 1 and 4, in the Lucas. Hmm? Well, let's do the Lucas swap elsewhere in the Fibonacci to be a bit more convincing. What about from 13, -8 to -8, 13 (and get a different sequence because we're keeping to the same addition rule)?

. . . -8 13 5 18 23 41 64 85 169

Do you think this result escapes the charge of triviality that I felt bound to level at my first attempt to hit the square? Do you feel it's twice as lovely? I do for the moment, but am always ready to reconsider.

## Hitting those squares - with some help from Monsieur Lucas

Hi Rachel. This is to continue our interesting discussion about a result with squares I'd thought I'd found in the Fibonacci sequence (see below). This time compare some of the sequence to the left as well as the right of 0 with a corresponding segment of the famous Lucas sequence:

Fibonacci: . . . 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 15 21 34 . . .

and

Lucas: . . . 47 -29 18 -11 7 -4 3 -1 2 1 0 3 4 7 11 18 29 47 . . .

The Lucas can be seen as resulting from swapping round two consecutive Fibonacci terms, from 2, -1 to -1, 2 while retaining the same addition rule as Fibonacci, adding two consecutive numbers to get the third as you go right. To the right in the Lucas we now have not just one but two integer squares, those of -1 and 2, namely 1 and 4, in the Lucas. Hmm? Well, let's do the Lucas swap elsewhere in the Fibonacci to be a bit more convincing. What about from 13, -8 to -8, 13 (and get a different sequence because we're keeping to the same addition rule)?

. . . -8 13 5 18 23 41 64 85 169

Do you think this result escapes the charge of triviality that I felt bound to level at my first attempt to hit the square? Do you feel it's twice as lovely? I do for the moment, but am always ready to reconsider.

Chris G