OFFSET
0,2
COMMENTS
The rows of Sierpiński's triangle, read as numbers in binary representation, are products of distinct Fermat numbers, row 0 being the empty product. (See also the comment in A080176.)
Rows 1 to 31 are the binary representation of the 31 (2^5-1) nonempty products of distinct Fermat primes, giving the number of sides of constructible (with straightedge and compass) odd-sided polygons. - Daniel Forgues, Jun 21 2011
Sierpiński's triangles typically refer to any finite triangle with rows 0 to 2^n-1 so as to get complete triangles, with n at least 4 so as to show the fractal-like pattern of nested triangles. We may consider these finite Sierpiński's triangles as finite parts of "the" infinite Sierpiński's triangle, so to speak. - Daniel Forgues, Jun 22 2011
Also, binary representation of the n-th iteration of the "Rule 60" elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016
a(n) is the concatenation of the coefficients of (x+1)^n in GF(2)[x]. - Thomas Anton, Oct 04 2022
REFERENCES
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 353.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..200
Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46. (For Denton Hewgill's identity)
OEIS Wiki, Sierpinski's triangle
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, arXiv:1011.6083 [math.NT], 2010-2012; J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
FORMULA
From Daniel Forgues, Jun 20-21 2011: (Start)
In the following formulas, [...]_2 means converted to base 2.
a(n) = [Sum_{i=0..n} (binomial(n,i) mod 2) 2^i]_2, n >= 0.
From row n, 0 <= n <= 2^k - 1, k >= 0, being
a(n) = [Product_{i=0..k-1} (F_i)^(alpha_i)]_2, alpha_i in {0, 1},
where for k = 0, we get the empty product, i.e., 1, giving a(0) = 1,
we induce from the triangle that row 2^k + n, 0 <= n <= 2^k - 1, is
a(2^k + n) = a(n)*[F_k]_2, k >= 0.
Denton Hewgill's identity (cf. links):
a(n) = [Product_{i>=0} (F_i)^(floor(n/2^i) mod 2)]_2, F_i = 2^(2^i)+1.
a(0) = 1; a(n) = [Product_{i=0..floor(log_2(n))} (F_i)^(floor(n/2^i) mod 2)]_2, F_i = 2^(2^i)+1, n >= 1. (End)
From Vladimir Shevelev, Dec 26-27 2013: (Start)
Sum_{n>=0} 1/a(n)^r = Product_{k>=0} (1 + 1/(10^(2^k)+1)^r),
Sum_{n>=0} (-1)^A000120(n)/a(n)^r = Product_{k>=0} (1 - 1/(10^(2^k)+1)^r), where r > 0 is a real number.
In particular,
Sum_{n>=0} 1/a(n) = Product_{k>=0} (1 + 1/(10^(2^k)+1)) = 1.10182034...;
Sum_{n>=0} (-1)^A000120(n)/a(n) = 0.9;
a(2^n) = 10^(2^n) + 1, n >= 0.
Note that analogs of Stephan's limit formulas (see Shevelev link) reduce to the relations a(2^t*n+2^(t-1)) = 99*(10^(2^(t-1)+1))/(10^(2^(t-1))-1) * a(2^t*n+2^(t-1)-2), t >= 2. In particular, for t=2,3,4, we have the following formulas:
a(4*n+2) = 101*a(4*n);
a(8*n+4) = (10001/101)*a(8*n+2);
a(16*n+8) = (100000001/1010101)*(16*n+6), etc. (End)
From Tom Edgar, Oct 11 2015: (Start)
a(2*n+1) = 11*a(2*n).
a(n) = Product_{b_j != 0} a(2^j) where n = Sum_{j>=0} b_j*2^j is the binary representation of n.
(End)
EXAMPLE
From Daniel Forgues, Jun 20 2011: (Start)
Terms as products of distinct Fermat numbers in binary representation (Cf. A080176 comment) (Cf. Sierpiński's triangle on OEIS Wiki):
a(0) = 1 = (empty product);
a(1) = 11 = F_0;
a(2) = 101 = F_1;
a(3) = 1111 = 11*101 = F_0*F_1;
a(4) = 10001 = F_2;
a(5) = 110011 = 11*10001 = F_0*F_2;
a(6) = 1010101 = 101*10001 = F_1*F_2;
a(7) = 11111111 = 11*101*10001 = F_0*F_1*F_2. (End)
MAPLE
A006943 := proc(n) local k; add((binomial(n, k) mod 2)*10^k, k=0..n); end;
MATHEMATICA
f[n_] := FromDigits@ Mod[Binomial[n, Range[0, n]], 2]; Array[f, 17, 0] (* Robert G. Wilson v, Jun 26 2011 *)
PROG
(Python)
def A006943(n): return sum((bool(~n&n-k)^1)*10**k for k in range(n+1)) # Chai Wah Wu, May 03 2023
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Aug 21 2000
Edited by Daniel Forgues, Jun 20 2011
STATUS
approved