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A138061 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). 0
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Row sums are:

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

This uses the French sideways graph method as in:

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

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

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

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

(* substitution *)

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

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

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

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

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

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

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

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

nr = k /. x -> z;

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

z = x + I*y;

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

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

LINKS

Table of n, a(n) for n=1..80.

FORMULA

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

EXAMPLE

First zero omitted:

{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, 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, 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, 50},

{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}

MATHEMATICA

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}];

CROSSREFS

Cf. A103684.

Sequence in context: A084426 A255049 A187564 * A257257 A257251 A068555

Adjacent sequences:  A138058 A138059 A138060 * A138062 A138063 A138064

KEYWORD

nonn,uned,tabf

AUTHOR

Roger L. Bagula, May 02 2008

STATUS

approved

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Last modified December 4 22:42 EST 2021. Contains 349526 sequences. (Running on oeis4.)