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 A006943 Rows of Sierpiński's triangle (Pascal's triangle mod 2). (Formerly M4802) 8
 1, 11, 101, 1111, 10001, 110011, 1010101, 11111111, 100000001, 1100000011, 10100000101, 111100001111, 1000100010001, 11001100110011, 101010101010101, 1111111111111111, 10000000000000001, 110000000000000011 (list; graph; refs; listen; history; text; internal format)
 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 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 V. 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 *) CROSSREFS Cf. A080176 for Fermat numbers in binary representation. Cf. A001317 for the decimal representation of A006943. Cf. A249183. Sequence in context: A288825 A290111 A193707 * A073030 A290295 A209930 Adjacent sequences:  A006940 A006941 A006942 * A006944 A006945 A006946 KEYWORD nonn,easy,base AUTHOR EXTENSIONS More terms from James A. Sellers, Aug 21 2000 Edited by Daniel Forgues, Jun 20 2011 STATUS approved

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Last modified April 21 01:53 EDT 2021. Contains 343143 sequences. (Running on oeis4.)