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A006943
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Rows of Sierpinski's triangle (Pascal's triangle mod 2.)
(Formerly M4802)
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2
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1, 11, 101, 1111, 10001, 110011, 1010101, 11111111, 100000001, 1100000011, 10100000101, 111100001111, 1000100010001, 11001100110011, 101010101010101, 1111111111111111, 10000000000000001, 110000000000000011
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The rows of Sierpinski'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) non-empty products of distinct Fermat primes, giving the number of sides of constructible (with straightedge and compass) odd-sided polygons. - Daniel Forgues, June 21 2011
Sierpinski'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 Sierpinski's triangles as finite parts of "the" infinite Sierpinski's triangle, so to speak. - Daniel Forgues, June 22 2011
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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).
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LINKS
| OEIS Wiki, Sierpinski's triangle
Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation (Abstract). (for Denton Hewgill's identity)
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FORMULA
| [Daniel Forgues, June 20-21 2011] (Start)
In the following formulae, [...]_2 means converted to base 2.
a(n) = [sum_(i=0..n) (binom(n,i) mod 2) 2^i]_2, n >= 0.
From row n, 0 <= n <= 2^k - 1, k >= 0, being
a(n) = [prod_(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) = [prod_(i=0..infty) (F_i)^(floor(n/2^i) mod 2)]_2, F_i = 2^(2^i)+1.
a(0) = 1; a(n) = [prod_(i=0..floor(log_2(n))) (F_i)^(floor(n/2^i) mod 2)]_2, F_i = 2^(2^i)+1, n >= 1. (End)
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EXAMPLE
| Terms as products of distinct Fermat numbers in binary representation (Cf. A080176 comment) (Cf. Sierpinski'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;
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MAPLE
| A006943 := proc(n) local k; add((binomial(n, k) mod 2)*10^k, k=0..n); end;
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MATHEMATICA
| f[n_] := FromDigits@ Mod[Binomial[n, Range[0, n]], 2]; Array[f, 17, 0] (* Robert G. Wilson v, Jun 26 2011 *)
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CROSSREFS
| Cf. A080176 for Fermat numbers in binary representation.
Cf. A001317 for the decimal representation of A006943.
Sequence in context: A199306 A113978 A193707 * A073030 A203304 A097649
Adjacent sequences: A006940 A006941 A006942 * A006944 A006945 A006946
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 21 2000
Edited by Daniel Forgues (kephalopod(AT)gmail.com), Jun 20 2011
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