In the Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…
 

When we take any four terms in the sequence such as 1, 1, 2, 3 the product of the first and last and twice the product of the inner two give integral solutions for the adjacent and opposite sides of the Pythagorean (right-angled) triangle.

Thus in this case 3 * 1 = 3 and 2(1 * 2) = 4 are the relevant sides.

Also the hypotenuse i.e. in this case 5 is a term in the Fibonacci sequence.

A similar type relationship holds for the Lucas series

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 501,…

Again when we take four terms in the sequence such as 1, 3, 4, 7 the product of the first and last and twice the product of the inner two, also give two integral solutions for the adjacent and opposite sides of the Pythagorean (right-angled) triangle.

Thus in this case 7 * 1 = 7 and 2(3 * 4) = 24 are the relevant sides.

In this case the hypotenuse is the sum of two terms, T_k andT_{k + 2} in the Lucas Series.

In this case the hypotenuse = T₄ + T₆ = 7 + 18 = 25

Here T₄ is the last term of the 4 term sequence.

When T₅ is the last term in the four-term sequence, the hypotenuse = T₆ + T₈;

When T₆ is the last term in the four-term sequence, the hypotenuse = T₈ + T₁₀ etc

So we moving out one term with each subsequent four-term sequence to drive the relevant hypotenuse terms.

Thus when T₇ is the last term in the four-term sequence, the hypotenuse = T₁₀ + T₁₂

The four-term sequence = 7, 11, 18, 29

Therefore the adjacent side = 29 * 7 = 203;

The opposite side = 2(11 * 18) = 396

The hypotenuse 123 + 322 = 445

Incidentally, when one compares the product of the last two terms in a four-term sequence with the product of the inner two, and interesting result emerges in that the difference always has an absolute value of 5.

Thus with the first four, the outer product = 7 and the inner product = 12 so that the difference = 5.

Likewise with 7, 11, 18, 29 the outer product = 203 and the inner product = 198 again with a difference = 5; (the sign alternates with each successive sequence!)

What is remarkable is that the hypotenuse generated in the Lucas case is always divisble by 5 with the result representing a term in the Fobonacci sequence.
Thus for example 445 = 89 * 5. (Interestingly there is a fascinating connection here with the sum of the Fibonacci and Lucas decimal point sequences. As we know the sum of the point series for the Fibonacci terms = the reciprocal of 89 whereas the sum of the Lucas point terms = the reciprocal of (89*5)/6).

In the Fibonacci Sequence when we apply the same procedure the difference always has an absolute value of 1.

Thus for example with the first four-term sequence, the outer product = 3 and the inner product = 2 with a difference = 1.