A179542 - OEIS

%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