Neat dispatch! I had misguidedly spent time attempting to fit second-order linear recurrences to the original sequences, before reluctantly accepting that they are actually third-order --- JPG's solution explains how my recurrences always have extra factor E-1 ( E == shift operator). The third sequence (exponential growth with alternating signs, symmetric about the -1 term) satisfies the algebraic constraints for a hyperbolic Steiner chain (intersecting frontier circles) with frontier curvatures 0,2 . However, consecutive members of this chain would touch internally --- apparently impossible in real plane geometry! So the final paragraph of my screed should be refined thus: << Geometric features of a chain, using t^2 > 0 for real n --- (6) 0 < d < 3 elliptic: A,B disjoint, incl. annular, closed; -1 < d < 0 elliptic: A,B disjoint and internal; 3 < d < oo hyperbolic: A,B intersecting; oo < d < -1 hyperbolic: infeasible in real geometry; d = 3 parabolic: Pappus arbelos, A,B touching; d = -1 degenerate: one of A,B is a point; d = oo or d = 0 impossible: eg. [1,1,1,2] ; d = 0/0 incomplete: extremum at k_(5/2) , incl. A,B concentric.
Also, the (Soddy) sample chain for n = 3 should read << [kA, kB; k1, k2, k3] = [ -1, 3; 2, 3, 2 ] , passim.
Fred Lunnon On 8/21/14, J.P. Grossman <jpg@alum.mit.edu> wrote:
Let d(n) = f(n) - f(n-1). Re-writing the recurrence in terms of d(n) and simplifying yields
( d(n) + d(n-2) ) / d(n-1) = ( d(n-1) + d(n-3) ) / d(n-2)
so in fact ( d(n) + d(n-2) ) / d(n-1) = C, a constant, and this is really a simpler recurrence in disguise:
d(n) = C * d(n-1) - d(n-2)
As long as C is an integer, you're all set (in your examples C = 1, 2 and -6, respectively).
J.P.
On Wed, Aug 20, 2014 at 10:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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
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