login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A155688 A symmetrical triangle of polynomial coefficients that are von Koch like: b=1/4; p(x, n) = If[Mod[n, 4] == 2, (b*x - n/2)*p(x, n - 1), If[ Mod[n, 4] == 3, (x/2 - b*n + 1/2)*p(x, n - 1), If[ Mod[n, 4] == 0, (-b*x - n/2 + b)*p(x, n - 1), (x/2 + b*n)*p(x, n - 1)]]]; q(x,n)=(p(x,n)+x^n*(p(1/x,n))/b^n. 0
2, 3, 3, -2, -14, -2, 8, -17, -17, 8, -32, -9, 226, -9, -32, -148, -85, 737, 737, -85, -148, 1672, 404, -6199, -2842, -6199, 404, 1672, -8416, 1744, 36297, -12993, -12993, 36297, 1744, -8416, 126016, -15504, -532423, 54438, 202722, 54438 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are:

{2, 6, -18, -18, 144, 1008, -11088, 33264, -532224, -5854464, 111234816,...}. Using the IFS definition of Hans Lauweier, I made a polynomial product set with Substirutions:

x->x and y->n.

The fractal modulo four is:

a = Table[Expand[(1/ b^n)*CoefficientList[ExpandAll[p[x, n]], x] + Reverse[(1/b^n)* CoefficientList[ExpandAll[p[x, n]], x]]], {n, 0, 128}];

b0 = Table[If[m <= n, Mod[ a[[n]][[m]], 4], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];

ListDensityPlot[b0, Mesh -> False, Frame -> False, AspectRatio -> Automatic, ColorFunction -> Hue]

REFERENCES

Hans Lauweier, Fractals,Endlessly Repeated Geometrical Figures,Princeton University Press, Ne Jersey,1991,pages 98-99

LINKS

Table of n, a(n) for n=0..41.

FORMULA

b=1/4;

p(x, n) = If[Mod[n, 4] == 2, (b*x - n/2)*p(x, n - 1),

If[ Mod[n, 4] == 3, (x/2 - b*n + 1/2)*p(x, n - 1),

If[ Mod[n, 4] == 0, (-b*x - n/2 + b)*p(x, n - 1),

(x/2 + b*n)*p(x, n - 1)]]];

q(x,n)=(p(x,n)+x^n*(p(1/x,n))/b^n.

EXAMPLE

{2},

{3, 3},

{-2, -14, -2},

{8, -17, -17, 8},

{-32, -9, 226, -9, -32},

{-148, -85, 737, 737, -85, -148},

{1672, 404, -6199, -2842, -6199, 404, 1672},

{-8416, 1744, 36297, -12993, -12993, 36297, 1744, -8416},

{126016, -15504, -532423, 54438, 202722, 54438, -532423, -15504, 126016},

{1134032, 111936, -4799127, -523664, 1149591, 1149591, -523664, -4799127, 111936, 1134032},

{-22679968, -1098304, 95967468, 4727137, -20196266, -2205318, -20196266, 4727137, 95967468, -1098304, -22679968}

MATHEMATICA

Clear[p, x, n, b, a, b0]; b = 1/4;

p[x, 0] = 1; p[x, 1] = x/2 + b; p[x_, n_] := p[x, n] = If[Mod[n, 4] == 2, (b*x - n/2)*p[x, n - 1],

If[Mod[n, 4] == 3, (x/2 - b*n + 1/2)*p[x, n - 1],

If[Mod[n, 4] == 0, (-b*x - n/2 + b)*p[x, n - 1], (x/2 + b*n)*p[x, n - 1]]]];

Table[Expand[(1/b^n)*CoefficientList[ ExpandAll[p[x, n]], x] + Reverse[(1/b^n)*CoefficientList[ExpandAll[ p[x, n]], x]]], {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A153479 A153489 A153310 * A215490 A153592 A153878

Adjacent sequences:  A155685 A155686 A155687 * A155689 A155690 A155691

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Jan 24 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 15:31 EDT 2019. Contains 328101 sequences. (Running on oeis4.)