A138061 - OEIS

%I #2 Mar 30 2012 17:34:26

%S 2,2,6,2,6,12,2,6,12,4,2,6,12,4,5,12,2,6,12,4,5,12,7,16,27,2,6,12,4,5,

%T 12,7,16,27,10,22,36,52,2,6,12,4,5,12,7,16,27,10,22,36,52,14,30,48,68,

%U 18,2,6,12,4,5,12,7,16,27,10,22,36,52,14,30,48,68,18,19,40,63,88,23,24

%N This sequence is a triangular sequence formed by the substitution: ( French sideways graph) 1->1,2;2->3;3->4;4->1; as a Markov style substitution form. The result is the differential polynomial coefficient form. ( first zero omitted).

%C Row sums are:

%C {0, 2, 8, 20, 24, 41, 91, 211, 389, 696, 1307}

%C This uses the French sideways graph method as in:

%C A103684:the morphism f: 1->{1,2}, 2->{1,3}, 3->{3}.

%C These sequences in the polynomial form were created to see what the

%C fractal implicit pictures would look like and not for the sequences:

%C Clear[a, s, p, t, m, n, t, p, k]

%C (* substitution *)

%C s[1] = {1, 2}; s[2] = {3}; s[3] = {4}; s[4] = {1};

%C t[a_] := Flatten[s /(AT) a];

%C p[0] = {1}; p[1] = t[p[0]];

%C p[n_] := t[p[n - 1]];

%C a = Table[p[n], {n, 0, 12}];

%C k = Table[D[Apply[Plus, Table[

%C a[[n]][[m]]*x^(m - 1), {m, 1, Length[a[[n]]]}]], x], {n, 3, 13}];

%C Clear[x, y, a, b, f, z, p];

%C nr = k /. x -> z;

%C p[z_] = Apply[Times, nr];

%C z = x + I*y;

%C f[x_, y_] = Re[1/(p[z])];

%C ContourPlot[ f[x, y], {x, -1.61,1.61}, {y, -1.61, 1.61}, PlotPoints -> {300, 300}, ImageSize ->600, ColorFunction -> (Hue[2# ] &)]

%F ( French sideways graph) 1->1,2;2->3;3->4;4->1; Substitution->p(x,n); out_n,m=Coefficients(dp(x,n)/dx).

%e First zero omitted:

%e {2},

%e {2, 6},

%e {2, 6, 12},

%e {2, 6, 12, 4},

%e {2, 6, 12, 4, 5, 12},

%e {2, 6, 12, 4, 5, 12, 7, 16, 27},

%e {2, 6, 12, 4, 5, 12, 7, 16, 27, 10, 22, 36, 52},

%e {2, 6, 12, 4, 5, 12, 7, 16, 27, 10, 22, 36, 52, 14, 30, 48, 68, 18},

%e {2, 6, 12, 4, 5, 12, 7, 16, 27, 10, 22, 36, 52, 14, 30, 48, 68, 18, 19, 40, 63, 88, 23, 24, 50},

%e {2, 6, 12, 4, 5, 12, 7, 16, 27, 10, 22, 36, 52, 14, 30, 48, 68, 18, 19, 40, 63, 88, 23, 24, 50, 26, 54, 84, 116, 30, 31, 64, 33, 68, 105}

%t Clear[a, s, p, t, m, n] (* substitution *) s[1] = {1, 2}; s[2] = {3}; s[3] = {4}; s[4] = {1}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]; a = Table[p[n], {n, 0, 10}]; Flatten[a]; b = Table[CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]*x^(m - 1), {m, 1, Length[a[[n]]]}]], x], x], {n, 1, 11}]; Flatten[b] Table[Apply[Plus, CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]*x^(m - 1), {m, 1, Length[a[[n]]]}]], x], x]], {n, 1, 11}];

%Y Cf. A103684.

%K nonn,uned,tabf

%O 1,1

%A _Roger L. Bagula_, May 02 2008