The values are related to the diophantine equations x^2 = 3*y^2 + 1 and x^2 = 3*y^2 - 2 It is a little easier to explain in terms of (1 + sqrt(3)) / 2 but this also works for (sqrt(3) - 1) / 2. For n = even number, let f(n) = (1 + sqrt(3))^n / 2^(n/2) = (x(n) + y(n)) / 2^(n/2); the values will satisfy x^2 = 3*y^2 + 1 For n = odd number, let f(n) = (1 + sqrt(3))^n / 2^((n+1)/2) = (x(n) + y(n)) / 2^((n+1)/2); the values will satisfy x^2 = 3*y^2 - 2 x(n+2) = 2*x(n) + 3*y(n) y(n+2) = 1*x(n) + 2*y(n) You can quickly get to f(10) f(2) = (2 + sqrt(3))/2 f(4) = (7 + 4*sqrt(3))/4 f(6) = (26 + 15*sqrt(3))/8 f(8) = (97 + 56*sqrt(3))/16 f(10) = (362 + 209*sqrt(3))/32 Note 362^2 = 3*209^2 + 1
What is interesting about this formula is the decimal value after the exponentiation ^10 . The result is almost zero: 4.3160 e-5 or 0.00004316 . If you were to exponentiate even further, e.g. ^15 , the final result would be much smaller: 0.0000002835
@mrinaldas9614
Ай бұрын
It is easier and more intuitive to go sqaring upto the power 8 and then multiplying with the squared value.
@遠傳五華
9 күн бұрын
+1
@RyanLewis-Johnson-wq6xs
Ай бұрын
It’s in my head.
@soundsoflife9549
11 күн бұрын
What if you stuck it in a row 10x then multiplied pairs so you had 5 pairs, then no srt3 of div 2 ect..
Пікірлер: 17