20 Aug
2014
20 Aug
'14
8:10 p.m.
Consider the recurrence f(n) = f(n-3) + ( f(n-4) - f(n-1) ) * ( f(n-2) - f(n-1) ) / ( f(n-2) - f(n-3) ) . Initialised with integer 4-tuplets, this generates sequences which are variously periodic: 6, -3, 12, 36, 45, 30, 6, -3, 12, 36, 45, 30, 6, -3, 12, 36, 45, 30, 6, ...; quadratic: 3, 1, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, ...; asymptotically exponential: 1, -1, 1, -9, 49, -289, 1681, -9801, 57121, -332929, 1940449, -11309769, ...; but remarkably remain integer despite the division operator. Explain this behaviour. WFL