Scale three Sierpinski triangle sequences

[Roger Bagula]

This result is at least one solution to the scale 3 Sierpinski
as triangular sequence.
I literally tried everythging I could think of to get something simple
that worked...
I had very bad luck in obtaining any good solution here.


%I A153516
%S A153516 2,3,3,2,14,2,2,25,25,2,2,33,92,33,2,2,41,200,200,41,2,2,49,340,676,340,
%T A153516 49,2,2,57,512,1616,1616,512,57,2,2,65,716,3148,5260,3148,716,65,2,2,73,
%U A153516 952,5400,13256,13256,5400,952,73,2
%N A153516 Triangular sequence recursion with row sums 2*3^(n-1): A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + 3*A(n - 2, k - 1)
%C A153516 Row sums are:
%C A153516 {2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366,...}
%e A153516 {2},
%e A153516 {3, 3},
%e A153516 {2, 14, 2},
%e A153516 {2, 25, 25, 2},
%e A153516 {2, 33, 92, 33, 2},
%e A153516 {2, 41, 200, 200, 41, 2},
%e A153516 {2, 49, 340, 676, 340, 49, 2},
%e A153516 {2, 57, 512, 1616, 1616, 512, 57, 2},
%e A153516 {2, 65, 716, 3148, 5260, 3148, 716, 65, 2},
%e A153516 {2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2}
%t A153516 Clear[t, n, m, A, a]; A[2, 1] := A[2, 2] = 3;
%t A153516 A[3, 2] = 14; A[4, 2] = 25; A[4, 3] = 25;
%t A153516 A[n_, 1] := 2; A[n_, n_] := 2;
%t A153516 A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + 3*A[n - 2, k - 1];
%t A153516 Table[Table[A[n, m], {m, 1, n}], {n, 1, 10}] ;
%t A153516 Flatten[%]
%K A153516 nonn
%O A153516 0,1
%A A153516 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 28 2008




And the next day:



The first row sums for scale 3 Sierpinski like sets were ( 2^n Pascal
like) :
2*3^n
Here we have at the Eulerian numbers ( (n+1)! row sums)
equivalent level
(n+2)!
I was very lucky this morning with finding a
solution on thec second try.
I calculated it to n=8 last night ( just a set of symmetrical values)
and then, got an nth
generalization of that which worked.

I spend most of the last few weeks trying to get the first level working
so this was nice. Next level (MacMahan equivalent)
should be row sums of 3^n*(n+2)!? (equivalent to row sums of 2^n*(n+1)! ).


%I A153592
%S A153592
2,3,3,2,20,2,2,58,58,2,2,100,516,100,2,2,162,2356,2356,162,2,2,248,
%T A153592
6718,26384,6718,248,2,2,362,16038,165038,165038,16038,362,2,2,508,
%U A153592
34256,664772,2229724,664772,34256,508,2,2,690,67344,2142448,17747916
%N A153592 A simple recursive with row sums (m+2)!;m=n-1: A(n,k)=A(n -
1, k - 1) + A(n - 1, k) + n*(n - 1)*A(n - 2, k - 1). %C A153592 Row sums:
%C A153592 {2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800,...}.
%C A153592 This recursion is Eulerian numbers equivalent level for a
scale three
%C A153592 Sierpinski set -Pascal triangular sequence( the sponge Bob
set). %F A153592 A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + n*(n - 1)*A(n -
2, k - 1). %e A153592 {2}, {3, 3}, {2, 20, 2}, {2, 58, 58, 2}, {2, 100,
516, 100, 2}, {2, 162, 2356, 2356, 162, 2}, {2, 248, 6718, 26384, 6718,
248, 2}, {2, 362, 16038, 165038, 165038, 16038, 362, 2}, {2, 508, 34256,
664772, 2229724, 664772, 34256, 508, 2}, {2, 690, 67344, 2142448,
17747916, 17747916, 2142448, 67344, 690, 2} %t A153592 Clear[A]; A[2, 1]
:= A[2, 2] = 3; A[3, 2] = 20;
%t A153592 A[4, 2] = 58; A[4, 3] = 58;
%t A153592 A[n_, 1] := 2; A[n_, n_] := 2;
%t A153592 A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + n*(n - 1)*A[n -
2, k - 1];
%t A153592 a = Table[A[n, k], {n, 10}, {k, n}];
%t A153592 Flatten[a] Table[Apply[Plus, a[[n]]], {n, 1, 10}];
Table[Apply[Plus, a[[n]]]/(n + 1)!, {n, 1, 10}]; %K A153592 nonn
%O A153592 1,1
%A A153592 Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 29 2008




I've been working of Sierpinski sets for more than ten years now.

Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net

[Roger Bagula]

A sample calculation of the next MacMahasn level
of scale three row sum triangular sequence:
%I A153637
%S A153637
2,9,9,2,212,2,2,1618,1618,2,2,2100,54116,2100,2,2,2786,609572,609572,
%T A153637
2786,2,2,3712,1582558,26220736,1582558,3712,2,2,4914,3257870,393546494,
%U A153637 393546494,3257870,4914,2,2,6428,6069056,1593218212,20609969404
%N A153637 A triangular sequence with row sums (3^(n - 1)*(n + 1)!)
starting at n=1 which was calculated by steps.
%C A153637 Row sums are:
%C A153637 {2, 18, 216, 3240, 58320, 1224720, 29393280, 793618560,
23808556800,
%C A153637 785682374400, 28284565478400}.
%e A153637 {2},
%e A153637 {9, 9},
%e A153637 {2, 212, 2},
%e A153637 {2, 1618, 1618, 2},
%e A153637 {2, 2100, 54116, 2100, 2},
%e A153637 {2, 2786, 609572, 609572, 2786, 2},
%e A153637 {2, 3712, 1582558, 26220736, 1582558, 3712, 2},
%e A153637 {2, 4914, 3257870, 393546494, 393546494, 3257870, 4914, 2},
%e A153637 {2, 6428, 6069056, 1593218212, 20609969404, 1593218212,
6069056, 6428, 2},
%e A153637 {2, 8290, 10645504, 4629106368, 388201427036, 388201427036,
4629106368, 10645504, 8290, 2},
%e A153637 {2, 10536, 17866010, 11449232704, 2180421367268,
23900788525360, 2180421367268, 11449232704, 17866010, 10536, 2}
%t A153637 Clear[a]; a = {{2}, {9, 9}, {2, 212, 2}, {2, 1618, 1618, 2},
%t A153637 {2, 2100, 54116, 2100, 2}, {2, 2786, 609572, 609572, 2786, 2},
%t A153637 {2, 3712, 1582558, 26220736, 1582558, 3712, 2}, {2,4914,
3257870, 393546494, 393546494, 3257870, 4914, 2},
%t A153637 {2, 6428, 6069056, 1593218212, 20609969404, 1593218212,
6069056, 6428, 2},
%t A153637 {2, 8290, 10645504, 4629106368, 388201427036, 388201427036,
4629106368, 10645504, 8290, 2},
%t A153637 {2, 10536, 17866010, 11449232704, 2180421367268,
23900788525360,2180421367268, 11449232704, 17866010, 10536, 2}};
%t A153637 Flatten[a] Table[Apply[Plus, a[[n]]], {n, 1, Length[a]}];
%t A153637 Table[Apply[Plus, a[[n]]]/(3^(n - 1)*(n + 1)!), {n, 1,
Length[a]}];
%K A153637 nonn
%O A153637 1,1
%A A153637 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 29 2008

Post by Roger Bagula
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html

[Roger Bagula]

In terms of fractal theory none of this is new,
but in terms of combinatorial theory
it appears to have never been tried before.

Once I got the 3^n row sequence right
I knew that the 5^n had to be close to it: ( the 3rd prime level
where scale three is the second and the Pascal's triangle is the first 2^n)
Mathematica:
Clear[t, n, m, A, a]
j = 3;
A[2, 1] := A[2, 2] = Prime[j];
A[3, 2] = 2*Prime[j]^2 - 4;
A[4, 2] = A[4, 3] = Prime[j]^3 - 2;;
A[n_, 1] := 2
A[n_, n_] := 2
A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + j*Prime[j]*A[n - 2, k - 1]
Table[Table[A[n, m], {m, 1, n}], {n, 1, 10}]
Flatten[%]
Table[Sum[A[n, m], {m, 1, n}], {n, 1, 10}]
Table[Sum[A[n, m], {m, 1, n}]/(2*5^(n - 1)), {n, 1, 10}]
a = Table[Table[If[m <= n, If[Mod[A[
n, m], 5] == 0, 0, 1], 0], {m, 1, 10}], {n, 1, 10}]
ListDensityPlot[a, Mesh -> False, Axes -> False]

%I A153648
%S A153648 2,5,5,2,46,2,2,123,123,2,2,155,936,155,2,2,187,2936,2936,187,2,2,219,
%T A153648 5448,19912,5448,219,2,2,251,8472,69400,69400,8472,251,2,2,283,12008,
%U A153648 159592,437480,159592,12008,283,2,2,315,16056,298680,1638072,1638072
%N A153648 A row sum 5^n triangular recursion sequence:Prime[j]=5=scale; A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + j*Prime[j]*A(n - 2, k - 1).
%C A153648 Row sums are:
%C A153648 {2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250,...}.
%C A153648 Plot of the lowest level of the fractal is:
%C A153648 a = Table[Table[If[m <= n, If[Mod[A[n, m], 5] == 0, 0, 1], 0], {m, 1, 10}], {n, 1, 10}] ;
%C A153648 ListDensityPlot[a, Mesh -> False, Axes -> False]
%F A153648 A(n,k)= A(n - 1, k - 1) + A(n - 1, k) + j*Prime[j]*A(n - 2, k - 1).
%e A153648 {2},
%e A153648 {5, 5},
%e A153648 {2, 46, 2},
%e A153648 {2, 123, 123, 2},
%e A153648 {2, 155, 936, 155, 2},
%e A153648 {2, 187, 2936, 2936, 187, 2},
%e A153648 {2, 219, 5448, 19912, 5448, 219, 2},
%e A153648 {2, 251, 8472, 69400, 69400, 8472, 251, 2},
%e A153648 {2, 283, 12008, 159592, 437480, 159592, 12008, 283, 2},
%e A153648 {2, 315, 16056, 298680, 1638072, 1638072, 298680, 16056, 315, 2}
%t A153648 Clear[t, n, m, A, a]; j = 3;
%t A153648 A[2, 1] := A[2, 2] = Prime[j];
%t A153648 A[3, 2] = 2*Prime[j]^2 - 4;
%t A153648 A[4, 2] = A[4, 3] = Prime[j]^3 - 2;
%t A153648 A[n_, 1] := 2; A[n_, n_] := 2;
%t A153648 A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + j*Prime[j]*A[n - 2, k - 1];
%t A153648 Table[Table[A[n, m], {m, 1, n}], {n, 1, 10}] ;
%t A153648 Flatten[%] Table[Sum[A[n, m], {m, 1, n}], {n, 1, 10}] ;
%t A153648 Table[Sum[A[n, m], {m, 1, n}]/(2*5^(n - 1)), {n, 1, 10}]
%K A153648 nonn
%O A153648 1,1
%A A153648 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 30 2008

[Roger Bagula]

http://www.iop.org/EJ/abstract/0305-4470/21/8/030
Fractal sequences derived from the self-similar extensions of the
Sierpinski gasket

A Lakhtakia et al 1988 J. Phys. A: Math. Gen. 21 1925-1928 doi:
10.1088/0305-4470/21/8/030 Help


PDF (219 KB) | References | Articles citing this article



A Lakhtakia, R Messier, V K Varadan and V V Varadan
Dept. of Eng. Sci. and Mech., Pennsylvania State Univ., University Park,
PA, USA

Abstract. The well known Sierpinski gasket has been generalised by the
authors into the generalised Pascal-Sierpinski gaskets (GPSG) of orders
(K, L), where both K and L are >or=2. It has been shown here that
families of self-similar sequences can be derived from these extensions
of the Sierpinski gasket when K=2 and L is a prime number.

Print publication: Issue 8 (21 April 1988)

Post by Roger Bagula
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html