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A006943 Rows of Sierpiński's triangle (Pascal's triangle mod 2).
(Formerly M4802)
7
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 (Abstract). (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, 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

Index to Elementary Cellular Automata

Index entries for sequences related to cellular automata

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

N. J. A. Sloane

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 August 22 07:02 EDT 2017. Contains 290943 sequences.