Description of A007413 as the rows of a triangle: Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 02 2008 %I A007413 %S A007413 1, 1,2,3, 1,2,3,1,3,2, 1,2,3,1,3,2,1,2,3,2,1,3, 1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2, 1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,2,3,2,1,3,1,2,3, 2,1,2,3,1,3,2,1,3 %N A007413 The characteristic polynomial of the graph matrix is -x^3+x^2+2*x %C A007413 These sequences as triangular forms have lengths like the Pc Mandelbrot-Julia polynomial; that fact suggests that the levels might be projected as coefficient polynomials. That presents an whole new way of seeing substitution sequences: 1, 1+2*x+3*x^2 1+2*x+3*x^2+x^3+3x^3+2*x^5 etc. Which can be projected to the complex plane as: x->a+i*b and plotted implicitly. %D A007413 A. Thue. Ueber unendliche Zeichenreihe. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 7:1a22, 1906. %D A007413 http://south.rotol.ramk.fi/keranen/ias2002/NewAbelianSquare-FreeDT0L-LanguagesOver4Letters.nb %F A007413 M= {{1, 1, 1}, {1, 0, 1}, {0, 1, 0}}. %e A007413 Triangular sequence form: {1}, {1,2,3}, {1,2,3,1,3,2}, {1,2,3,1,3,2,1,2,3,2,1,3}, {1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2}, {1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,1,3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,2,3,2,1,3,1,2,3,2,1,2,3,1,3,2,1,3} %t A007413 Clear[productions, a] productions = {"1" -> "1,2,3", "2" -> "1,3", "3" -> "2", " " -> ""};; g[x_] := StringReplace[x, productions] a = NestList[g, "1", 5] %Y A007413 Cf. A001285