ScienceNews direct: Tell the world about your findings: Digital language of Pascal's triangle (Part 3)

Digital language of Pascal's triangle

Lutvo Kurić

(Part 3)

Independent Researcher

Bosnia and Herzegovina

72290 Novi Travnik

Kalinska 7

Tel. 061 763 917

lutvokuric@yahoo.com

Diagonals in Pascal’s triangle

·	·	·	·	·	·	·	·	· Sum
· D1	· 1	· 1	· 1	· 1	· 1	· 1	· 1.....	· 7
· D2	· 1	· 2	· 3	· 4	· 5	· 6	· 7.....	· 28
· D3	· 1	· 3	· 6	· 10	· 15	· 21	· 28....	· 84
· D4	· 1	· 4	· 10	· 20	· 35	· 56	· 84....	· 210
· D5	· 1	· 5	· 15	· 35	· 70	· 126	· 210....	· 462
· D6	· 1	· 6	· 21	· 56	· 126	· 252	· 462....	· 924
· D7	· 1	· 7	· 28	· 84	· 210	· 462	· 924....	· 1716
· D8	· 1	· 8	· 36	· 120	· 330	· 792	· 1716....	· 3003
· D9	· 1	· 9	· 45	· 165	· 495	· 1287	· 3003....	· 5005
· D10	· 1	· 10	· 55	· 220	· 715	· 2002	· 5005....	· 8008
· D11	· 1	· 11	· 66	· 286	· 1001	· 3003	· 8008....	· 12376
· D12	· 1	· 12	· 78	· 364	· 1365	· 4368	· 12376....	· 18564
· D13	· 1	· 13	· 91	· 455	· 1820	· 6188	· 18564....	· 27132
· D14	· 1	· 14	· 105	· 560	· 2380	· 8568	· 27132....	· 38760
· D15	· 1	· 15	· 120	· 680	· 3060	· 11628	· 38760....	· 54264
· D16	· 1	· 16	· 136	· 816	· 3876	· 15504	· 54264....	· 74613
· D17	· 1	· 17	· 153	· 969	· 4845	· 20349	· 74613....	· 100947
· D18	· 1	· 18	· 171	· 1140	· 5985	· 26334	· 100947....	· 134596
· D19	· 1	· 19	· 190	· 1330	· 7315	· 33649	· 134596....	· 177100
· D20	· 1	· 20	· 210	· 1540	· 8855	· 42504	· 177100....	· 230230
· D21	· 1	· 21	· 231	· 1771	· 10626	· 53130	· 230230....	· 296010
·	·	·	·	·	·	·	·	· Etc.

Diagonal D1

(1+1+1+1+1+1+1) = 7;

Diagonal D2

Input = Diagonal D1

Output = Diagonal D2

(1+1+1+1+1+1+1) è (1+(1+1)+(1+1+1)+(1+1+1+1)+(1+1+1+1+1)+(1+1+1+1+1+1)+

+ (1+1+1+1+1+1+1) = (1+2+3+4+5+6+7) =28:

Diagonal D3

Input = Diagonal D2

Output = Diagonal D3

(1+2+3+4+5+6+7) è (1+(1+2)+(1+2+3)+(1+2+3+4)+(1+2+3+4+5)+ +(1+2+3+4+5+6)+(1+2+3+4+5+6+7) = (1+3+6+10+15+21+28) = 84;

Diagonal D4

Input = Diagonal D3

Output = Diagonal D4

(1+3+6+10+15+21+28) è (1+(1+3)+(1+3+6)+(1+3+6+10)+(1+3+6+10+15) +

+ (1+3+6+10+15+21) + (1+3+6+10+15 + +21+28) = (1+4+10+20+35+56+84)= 210:

Diagonal D5

Input = Diagonal D4

Output = Diagonal D5

(1+4+10+20+35+56+84) è (1+(1+4)+(1+4+10)+(1+4+10+20)+(1+4+10+20+35)+

+(1+4+10+20+35+56)+(1+4+10+20+35+56+84) = (1+5+15+35+70+126+210) = 462;

etc.

Row 1 = 1,3,6,10,15,21,28;

Row 2, column 1 = 1;

Row 2, column 2 = (Row 1, column 1 + row 2) = (1+3) = 4;

Row 2, column 3 = (Row 2, column 2 + row 1, column 3) = (4+6) = 10;

Row 2, column 4 = (Row 2, column 3 + row 1, column 4) = (10+10) = 20;

etc.

A polynomial expression with three terms, such as:

A0 è1; 1 = 10101⁰

A1 è1,01,01; 10101 = 10101¹

A2 è1,02,03,02,01; 102030201 = 10101²

A3 è 1,03,06,07,06,03,01; 1030607060301= 10101³

A4 è 1,04,10,16,19,16,10,04,01; 10410161916100401= 10101⁴

A5 è1,05,15,30,45,51,45,30,15,05,01; 105153045514530150501= 10101⁵

^etc.

Groups of numbers

Row Number	Binomial Expansion
0	1
1	2
2	4
3	8
4	16
5	32
6	64
7	128
N	Y

Groups with 2 numbers

Example 1

Row Number	Binomial Expansion
0	1
1	2
Sum	3

Example 2

Row Number	Binomial Expansion
1	2
2	4
Sum	3 x 2

Example 3

Row Number	Binomial Expansion
2	4
3	8
Sum	3 x 4

Example "n"

Row Number	Binomial Expansion
X1	Y1
X2	Y2
Sum	3 x P

P = (1,2,4,8,16,32,64,128, etc.).

Gr(2) = (1+2) = 3;

Groups with 3 numbers

Example 1

Row Number	Binomial Expansion
0	1
1	2
2	4
Sum	7

Example 2

Row Number	Binomial Expansion
1	2
2	4
3	8
Sum	7 x 2

Example 3

Row Number	Binomial Expansion
2	4
3	8
4	16
Sum	7 x 4

Example "n"

Row Number	Binomial Expansion
X1	Y1
X2	Y2
X3	Y3
Sum	7 x P

P = (1,2,4,8,16,32,64,128, etc.).

Groups with 4 numbers

Example 1

Row Number	Binomial Expansion
0	1
1	2
2	4
3	8
Sum	15

Example 2

Row Number	Binomial Expansion
1	2
2	4
3	8
4	16
Sum	15 x 2

Example 3

Row Number	Binomial Expansion
2	4
3	8
4	16
5	32
Sum	15 x 4

Example "n"

Row Number	Binomial Expansion
X1	Y1
X2	Y2
X3	Y3
X4	Y4
Sum	15 x P

P = (1,2,4,8,16,32,64,128, etc.).

Gr(4) = 15;

etc.

Arithmetical expression for binomial expansion in these examples are number, 1, 3, 7, 15, etc.

Code 11

The powers of 11 can be extracted from Pascal's triangle by reading across the rows and interpreting the digits as a place value system.. We can think of row 1-8 in this way:

Row 1 = 11;

Row 2 = 121 = (11x11);

Row 3 = 1331 = (11x11x11);

Row 4 = 14641 = (11x11x11x11);

(2,01) – (1,02) è (201-102) =99 = (11 x Y);

(3, 01) – (1,03) è (301-103) = (11 x Y1);

(6,03,01) – (1,03,06) è (11 x Y2);

(10,06,03,01) – (1,03,06,10) è (11 x Y3);

(15,10,06,03,01) – (1,03,06,10,15) è (11 x Y4);

(21,15,10,06,03,01) – (1,03,06,10,15,21) è (11 x Y5);

(28,21,15,10,06,03,01) – (1,03,06,10,15,21,28) è (11 x Y6);

(36,28,21,15,10,06,03,01) – (1,03,06,10,15,21,28,36) è (11 x Y7);

(10,04,01) – (1,04,10) è (11 x Y8)

(35,20,10,04,01) – (1,04,10,20,35) è (11 x Y9)

(462,252,126,056,021,006,001) – (1,006,021,056,126,252,462) è (11 x Y10);

etc.

Code 7

								1
							1		1
						1		2		1
					1		3		3		1
				1		4		6		4		1
			1		5		10		10		5		1
		1		6		15		20		15		6		1
	1		7		21		35		35		21		7		1
1		8		28		56		70		56		28		8		1

[1+ (1+1) + (1+2+1)] = (7 x 1);

[(1+1) + (1+2+1) + (1+3+3+1)] = (7 x 2);

[(1+2+1) + (1+3+3+1) + (1+4+6+4+1)] = (7 x 4);

[(1+3+3+1) + (1+4+6+4+1) + (1+5+10+10+5+1)] = (7 x 8);

[(1+4+6+4+1) + (1+5+10+10+5+1) + (1+6+15+20+15+6+1)] = (7 x 16);

[(1+5+10+10+5+1) + (1+6+15+20+15+6+1) + (1+7+21+35+35+21+7+1)] = (7 x 32);

[(1+6+15+20+15+6+1) + (1+7+21+35+35+21+7+1) +

+ (1+8+28+56+70+56+28+8+1)] = (7 x 64);

[(1 + (1+2+1) + (1+4+6+4+1)] = (7 x 3);

[(1+2+1) + (1+4+6+4+1) + (1+6+15+20+15+6+1)] = (7 x 12);

[(1+4+6+4+1) + (1+6+15+20+15+6+1) + (1+8+28+56+70+56+28+8+1)] = (7 x 48);

etc.

CONCLUSION:

Making a sequence of all numbers in Pascal triangle and Binomial Expansion is conducted according to the exact cybernetic laws (for such descriptions we can use theory of systems programs, informations and cybernetics.)

Literature:

The Math Forum
http://mathforum.org/

Labels: Lutvo Kurić