A179542 Trajectory of 1 under the morphism 1->(1,2,3), 2->(1,2), 3->(1) related to the heptagon and A006356.

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, 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, 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

Offset: 0

Author

Gary W. Adamson, Jul 18 2010

Keywords

• nonn

Comments

Given M = the generating matrix for the heptagon shown in A006356:

[1,1,1; 1,1,0; 1,0,0] take powers of M, extracting top row getting:

(1,1,1), (3,2,1), (6,5,3), (14,11,6), where left and right columns (offset) =

A006356, and middle column = A006054. n-th iterate of the sequence is

composed of A006356(n) terms parsed into a frequency of 1's, 2's, and 3's

matching the 3-termed vectors with appropriate sums.

Examples

			Starting with 1, the next two iterates are:
(1, 2, 3) -> (1, 2, 3, 1, 2, 1) -> (1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3).
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).
		

Links

• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A006356, A006054

Programs

Mathematica

NestList[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 2}, 3 -> 1}] &, {1}, 5] // Flatten (* Robert G. Wilson v, Jul 23 2010 *)

Extensions

More terms from Robert G. Wilson v, Jul 23 2010