new general triangle form : w=2 is Pascal
- To: mathgroup at smc.vnet.net
- Subject: [mg105501] new general triangle form : w=2 is Pascal
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Sat, 5 Dec 2009 05:34:28 -0500 (EST)
- Reply-to: rlbagulatftn at yahoo.com
I've already posted this to several of my yahoo
groups:
They alternate odd even with the modulo two stagger Sierpinski
and Sierpinski fractals:
The general form links three known sequences so far.
It is always nice to find a new Sierpinski-Pascal type.
These sequences have very low coefficients compared to the Eulerian numbers
and MacMahon numbers type sequence.
w=0
%S A051159
1,1,1,1,0,1,1,1,1,1,1,0,2,0,1,1,1,2,2,1,1,1,0,3,0,3,0,1,1,1,3,3,3,3,1,
%T A051159
1,1,0,4,0,6,0,4,0,1,1,1,4,4,6,6,4,4,1,1,1,0,5,0,10,0,10,0,5,0,1,1,1,5,
%U A051159 5,10,10,10,10,5,5,1,1
%E A051159 A new way to calculate the sequence is presented. Roger L.
Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009
%C A051159 Contribution from Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%C A051159 The sequence of coefficients of a general polynomial
recursion that links at w=2 to the Pascal triangle is here w=0.
%C A051159 Row sums are:
%C A051159 {1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64,...} (End)
%F A051159 w-0:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 +
w*x + 1)^Floor[n/2]] [From Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 04 2009]
%e A051159 Contribution from Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%e A051159 {1},
%e A051159 {1, 1},
%e A051159 {1, 0, 1},
%e A051159 {1, 1, 1, 1},
%e A051159 {1, 0, 2, 0, 1},
%e A051159 {1, 1, 2, 2, 1, 1},
%e A051159 {1, 0, 3, 0, 3, 0, 1},
%e A051159 {1, 1, 3, 3, 3, 3, 1, 1},
%e A051159 {1, 0, 4, 0, 6, 0, 4, 0, 1},
%e A051159 {1, 1, 4, 4, 6, 6, 4, 4, 1, 1},
%e A051159 {1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1},
%e A051159 {1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1} (End)
%t A051159 Contribution from Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%t A051159 Clear[p, n, x, a]
%t A051159 w = 0;
%t A051159 p[x, 1] := 1;
%t A051159 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n -
1], (x^2 + w*x + 1)^Floor[n/2]]
%t A051159 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
%t A051159 Flatten[a] (End)
%Y A051159 A169623 [From Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Dec 04 2009]
w=1
%I A169623
%S A169623 1,1,1,1,1,1,1,2,2,1,1,2,3,2,1,1,3,5,5,3,1,1,3,6,7,6,3,1,1,4,9,13,13,9,
%T A169623 4,1,1,4,10,16,19,16,10,4,1,1,5,14,26,35,35,26,14,5,1,1,5,15,30,45,51,
%U A169623 45,30,15,5,1,1,6,20,45,75,96,96,75,45,20,6,1
%N A169623 Coefficients of polynomials: p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x2 + x + 1)^Floor[n/2]]
%C A169623 Row sums are:A038754;
%C A169623 {1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486,...}. The modulo two fractal given by these polynomials is non-Sierpinski:
%C A169623 call it stagger Sierpinski.
%C A169623 a = Table[CoefficientList[p[x, n], x], {n, 1, 128}];
%C A169623 ListDensityPlot[Table[If[m <= n,Mod[a[[n, m]], 2], 0], {m, 1, 128}, {n, 1, 128}], Mesh -> False, Frame -> False]
%F A169623 p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x2 + x + 1)^Floor[n/2]]
%e A169623 {1},
%e A169623 {1, 1},
%e A169623 {1, 1, 1},
%e A169623 {1, 2, 2, 1},
%e A169623 {1, 2, 3, 2, 1},
%e A169623 {1, 3, 5, 5, 3, 1},
%e A169623 {1, 3, 6, 7, 6, 3, 1},
%e A169623 {1, 4, 9, 13, 13, 9, 4, 1},
%e A169623 {1, 4, 10, 16, 19, 16, 10, 4, 1},
%e A169623 {1, 5, 14, 26, 35, 35, 26, 14, 5, 1},
%e A169623 {1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1},
%e A169623 {1, 6, 20, 45, 75, 96, 96, 75, 45, 20, 6, 1}
%t A169623 Clear[p, n, x, a]
%t A169623 p[x, 1] := 1;
%t A169623 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x2 + x + 1)^Floor[n/2]]
%t A169623 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
%t A169623 Flatten[a]
%Y A169623 A038754
%K A169623 nonn
%O A169623 1,8
%A A169623 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 03 2009
w=2 ->Pascal A007318
w=3
%S A026374 1,1,1,1,0,1,1,1,1,1,1,0,2,0,1,1,1,2,2,1,1,1,0,3,0,3,0,1,1,1,3,3,3,3,1,
%T A026374 1,1,0,4,0,6,0,4,0,1,1,1,4,4,6,6,4,4,1,1,1,0,5,0,10,0,10,0,5,0,1,1,1,5,
%U A026374 5,10,10,10,10,5,5,1,1
%E A026374 A new way to calculate the sequence is presented. Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009
%C A026374 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%C A026374 The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=3.
%C A026374 Row sums are:
%C A026374 {1, 2, 5, 10, 25, 50, 125, 250, 625, 1250, 3125, 6250,...} (End)
%F A026374 w=3:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009]
%e A026374 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%e A026374 {1},
%e A026374 {1, 1},
%e A026374 {1, 3, 1},
%e A026374 {1, 4, 4, 1},
%e A026374 {1, 6, 11, 6, 1},
%e A026374 {1, 7, 17, 17, 7, 1},
%e A026374 {1, 9, 30, 45, 30, 9, 1},
%e A026374 {1, 10, 39, 75, 75, 39, 10, 1},
%e A026374 {1, 12, 58, 144, 195, 144, 58, 12, 1},
%e A026374 {1, 13, 70, 202, 339, 339, 202, 70, 13, 1},
%e A026374 {1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1},
%e A026374 {1, 16, 110, 425, 1015, 1558, 1558, 1015, 425, 110, 16, 1} (End)
%t A026374 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)
%t A026374 Clear[p, n, x, a]
%t A026374 w = 0;
%t A026374 p[x, 1] := 1;
%t A026374 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]]
%t A026374 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
%t A026374 Flatten[a] (End)
%Y A026374 A051159 ,A169623, A007318 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009]
w=4
%I A171142
%S A171142 1,1,1,1,4,1,1,5,5,1,1,8,18,8,1,1,9,26,26,9,1,1,12,51,88,51,12,1,1,13,
%T A171142 63,139,139,63,13,1,1,16,100,304,454,304,100,16,1,1,17,116,404,758,758,
%U A171142 404,116,17,1,1,20,165,720,1770,2424,1770,720,165,20,1,1,21,185,885
%N A171142 The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=4:p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%C A171142 Row sums are:
%C A171142 {1, 2, 6, 12, 36, 72, 216, 432, 1296, 2592, 7776, 15552,...}
%F A171142 w=4;
%F A171142 p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%e A171142 {1},
%e A171142 {1, 1},
%e A171142 {1, 4, 1},
%e A171142 {1, 5, 5, 1},
%e A171142 {1, 8, 18, 8, 1},
%e A171142 {1, 9, 26, 26, 9, 1},
%e A171142 {1, 12, 51, 88, 51, 12, 1},
%e A171142 {1, 13, 63, 139, 139, 63, 13, 1},
%e A171142 {1, 16, 100, 304, 454, 304, 100, 16, 1},
%e A171142 {1, 17, 116, 404, 758, 758, 404, 116, 17, 1},
%e A171142 {1, 20, 165, 720, 1770, 2424, 1770, 720, 165, 20, 1},
%e A171142 {1, 21, 185, 885, 2490, 4194, 4194, 2490, 885, 185, 21, 1}
%t A171142 Clear[p, n, x, a]
%t A171142 w = 4;
%t A171142 p[x, 1] := 1;
%t A171142 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]];
%t A171142 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
%t A171142 Flatten[a]
%Y A171142 A051159 ,A169623, A007318
%K A171142 nonn
%O A171142 1,5
%A A171142 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009
w=5
%I A171143
%S A171143 1,1,1,1,5,1,1,6,6,1,1,10,27,10,1,1,11,37,37,11,1,1,15,78,155,78,15,1,1,
%T A171143 16,93,233,233,93,16,1,1,20,154,560,931,560,154,20,1,1,21,174,714,1491,
%U A171143 1491,714,174,21,1,1,25,255,1350,3885,5775,3885,1350,255,25,1,1,26,280
%N A171143 The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=5:p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%C A171143 Row sums are:
%C A171143 {1, 2, 7, 14, 49, 98, 343, 686, 2401, 4802, 16807, 33614,...}
%F A171143 w=5;
%F A171143 p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%e A171143 {1},
%e A171143 {1, 1},
%e A171143 {1, 5, 1},
%e A171143 {1, 6, 6, 1},
%e A171143 {1, 10, 27, 10, 1},
%e A171143 {1, 11, 37, 37, 11, 1},
%e A171143 {1, 15, 78, 155, 78, 15, 1},
%e A171143 {1, 16, 93, 233, 233, 93, 16, 1},
%e A171143 {1, 20, 154, 560, 931, 560, 154, 20, 1},
%e A171143 {1, 21, 174, 714, 1491, 1491, 714, 174, 21, 1},
%e A171143 {1, 25, 255, 1350, 3885, 5775, 3885, 1350, 255, 25, 1},
%e A171143 {1, 26, 280, 1605, 5235, 9660, 9660, 5235, 1605, 280, 26, 1}
%t A171143 Clear[p, n, x, a]
%t A171143 w = 5;
%t A171143 p[x, 1] := 1;
%t A171143 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]];
%t A171143 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
%t A171143 Flatten[a]
%Y A171143 A051159 ,A169623, A007318,A171142
%K A171143 nonn
%O A171143 1,5
%A A171143 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009
w=6
%I A171144
%S A171144 1,1,1,1,6,1,1,7,7,1,1,12,38,12,1,1,13,50,50,13,1,1,18,111,252,111,18,1,
%T A171144 1,19,129,363,363,129,19,1,1,24,220,936,1734,936,220,24,1,1,25,244,1156,
%U A171144 2670,2670,1156,244,25,1,1,30,365,2280,7570,12276,7570,2280,365,30,1,1
%N A171144 The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=6:p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%C A171144 Row sums are:
%C A171144 {1, 2, 8, 16, 64, 128, 512, 1024, 4096, 8192, 32768, 65536,...}
%F A171144 w=6;
%F A171144 p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]]
%e A171144 {1},
%e A171144 {1, 1},
%e A171144 {1, 6, 1},
%e A171144 {1, 7, 7, 1},
%e A171144 {1, 12, 38, 12, 1},
%e A171144 {1, 13, 50, 50, 13, 1},
%e A171144 {1, 18, 111, 252, 111, 18, 1},
%e A171144 {1, 19, 129, 363, 363, 129, 19, 1},
%e A171144 {1, 24, 220, 936, 1734, 936, 220, 24, 1},
%e A171144 {1, 25, 244, 1156, 2670, 2670, 1156, 244, 25, 1},
%e A171144 {1, 30, 365, 2280, 7570, 12276, 7570, 2280, 365, 30, 1},
%e A171144 {1, 31, 395, 2645, 9850, 19846, 19846, 9850, 2645, 395, 31, 1}
%t A171144 Clear[p, n, x, a]
%t A171144 w = 6;
%t A171144 p[x, 1] := 1;
%t A171144 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]];
%t A171144 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
%t A171144 Flatten[a]
%Y A171144 A051159 ,A169623, A007318,A171142,A171143
%K A171144 nonn
%O A171144 1,5
%A A171144 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009
They keep on going ,but that is enough to establish
the general form.
Respectfully, Roger L. Bagula
11759 Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
http://www.google.com/profiles/Roger.Bagula
alternative email: roger.bagula at gmail.com