Random bits of knowledge

Fibonacci

${matrix{3}{1} { {F_1 = 0} {F_2 = 1} {F_n = F_{n-1} + F_{n-2}} }}$

${matrix{2}{1} { {varphi={1+sqrt 5}/2} {F_{n}={varphi^n-( -varphi)^-n}/sqrt 5} }}$

${matrix{3}{1} { {F_{2n} = {F_n}^2+{F_{n+1}}^2} {F_{2n+1} = (2F_n+F_{n+1})F_{n+1}} {F_{2n-1} = (2F_{n+1}-F_n)F_n} } }$

${matrix{2}{2} { {for n ulige}{F_{n} = {(F_{{n-1}/2}})^2+{(F_{{n+1}/2}})^2} {for n lige}{F_{n} = (2F_{n/2-1}+F_{n/2})F_{n/2}} } }$

Formula til non-Fibonacci tal: $floor ( n + log_varphi (sqrt 5(log_varphi(n sqrt{5})+n)−5+{3}/n)−2 ) for n>=2$