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%I #12 Sep 19 2017 04:29:17
%S 1,1,2,3,1,2,3,1,2,1,1,2,3,1,2,1,1,2,3,1,2,1,2,3,1,2,3,1,2,1,1,2,3,1,
%T 2,1,2,3,1,2,3,1,2,1,1,2,3,1,2,1,2,3,1,2,1,1,2,3,1,2,1,1,2,3,1,2,1,2,
%U 3,1,2,3,1,2,1,1,2,3,1,2,1,2,3,1,2,1,1,2,3,1,2,1,1,2,3,1,2,1,2,3,1,2,3,1,2
%N Trajectory of 1 under the morphism 1->(1,2,3), 2->(1,2), 3->(1) related to the heptagon and A006356.
%C Given M = the generating matrix for the heptagon shown in A006356:
%C [1,1,1; 1,1,0; 1,0,0] take powers of M, extracting top row getting:
%C (1,1,1), (3,2,1), (6,5,3), (14,11,6), where left and right columns (offset) =
%C A006356, and middle column = A006054. n-th iterate of the sequence is
%C composed of A006356(n) terms parsed into a frequency of 1's, 2's, and 3's
%C matching the 3-termed vectors with appropriate sums.
%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%e Starting with 1, the next two iterates are:
%e (1, 2, 3) -> (1, 2, 3, 1, 2, 1) -> (1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3).
%e The 3rd iterate has 14 terms composed of six 1's, five 2's, and three 3's; matching the top row of M^3 = (6, 5, 3), sum = 14 = A006356(3).
%t NestList[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 2}, 3 -> 1}] &, {1}, 5] // Flatten (* _Robert G. Wilson v_, Jul 23 2010 *)
%Y Cf. A006356, A006054
%K nonn
%O 0,3
%A _Gary W. Adamson_, Jul 18 2010
%E More terms from _Robert G. Wilson v_, Jul 23 2010
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