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Sierpiński's triangle
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Sierpiński's triangle, named after Wacław Sierpiński, is Pascal's triangle modulo 2.
Contents
2 
n 
[row(n)]10 
i = 
Sierpiński's triangle rows



Factorization into Fermat numbers  

0  1  1  
1  11  3 
 
2  101  5 
 
3  1111  15 
 
4  10001  17 
 
5  110011  51 
 
6  1010101  85 
 
7  11111111  255 
 
8  100000001  257 
 
9  1100000011  771 
 
10  10100000101  1285 
 
11  111100001111  3855 
 
12  1000100010001  4369 
 
13  11001100110011  13107 
 
14  101010101010101  21845 
 
15  1111111111111111  65535 
 
16  10000000000000001  65537 
 
17  110000000000000011  196611 
 
18  1010000000000000101  327685 
 
19  11110000000000001111  983055 
 
20  100010000000000010001  1114129 
 
21  1100110000000000110011  3342387 
 
22  10101010000000001010101  5570645 
 
23  111111110000000011111111  16711935 
 
24  1000000010000000100000001  16843009 
 
25  11000000110000001100000011  50529027 
 
26  101000001010000010100000101  84215045 
 
27  1111000011110000111100001111  252645135 
 
28  10001000100010001000100010001  286331153 
 
29  110011001100110011001100110011  858993459 
 
30  1010101010101010101010101010101  1431655765 
 
31  11111111111111111111111111111111  4294967295 
 
32  100000000000000000000000000000001  4294967297* 

F5 = 4294967297 = 641 ⋅ 6700417 
Sierpinski's triangle rows and constructible (with straightedge and compass) oddsided polygons
Forn = 1 
31 
Rows formulae
It can be proved by induction that the rows of Sierpinski's triangle interpreted as a binary number are products of distinct Fermat numbers (Denton Hewgill's identity^{[1]})
Inductive proof:
 Base case: For row 0, we get 1 (empty product), the product of distinct Fermat numbers in { };
 Inductive hypothesis: Assume that any row
is the product of distinct Fermat numbers inn, 0 ≤ n ≤ 2 k − 1, {F0, ..., Fk − 1}
2 k + n, 0 ≤ n ≤ 2 k − 1, 
2 n, n ≥ 0, 
Fn 
n 
Sequences
A006943 Sierpinski's triangle (Pascal's triangle mod 2) rows.

{1, 11, 101, 1111, 10001, 110011, 1010101, 11111111, 100000001, 1100000011, 10100000101, 111100001111, 1000100010001, 11001100110011, 101010101010101, 1111111111111111, 10000000000000001, 110000000000000011, 1010000000000000101, ...}
A047999 Sierpinski's triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle mod 2.

{1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, ...}
n ≥ 0. 

{1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, ...}
See also
Notes
 ↑ Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation (Abstract).
External links
 Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, The Fibonacci Quarterly, Volume 42, Number 1, February 2004.