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A038183 One-dimensional cellular automaton 'sigma-minus' (Rule 90): 000,001,010,011,100,101,110,111 -> 0,1,0,1,1,0,1,0. 19
1, 5, 17, 85, 257, 1285, 4369, 21845, 65537, 327685, 1114129, 5570645, 16843009, 84215045, 286331153, 1431655765, 4294967297, 21474836485, 73014444049, 365072220245, 1103806595329, 5519032976645, 18764712120593, 93823560602965 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generation n (starting from the generation 0: 1) interpreted as a binary number.

Observation: for n <= 15, a(n) = smallest number whose Euler totient is divisible by 4^n. This is not true for n = 16. - Arkadiusz Wesolowski, Jul 29 2012

REFERENCES

Stephen Wolfram: Geometry of Binomial Coefficients, Amer. Math. Monthly, Volume 91, Number 9, November 1984, pages 566-571.

LINKS

Table of n, a(n) for n=0..23.

S. Wolfram, O. Martin, and A.M. Odlyzko, Algebraic Properties of Cellular Automata (1984), Communications in Mathematical Physics, 93 (March 1984) 219-258.

Eric Weisstein's World of Mathematics, Rule 90

Index entries for sequences related to cellular automata

Wikipedia, Rule 90

FORMULA

a(n) = Product(((bit_n(n, i)*(2^(2^(i+1))))+1), i=0..inf); # A direct algebraic formula!

a(n)=sum{k=0..n, (C(2*n, 2*k) mod 2)*4^(n-k)}. - Paul Barry, Jan 03 2005

a(2*n+1) = 5*a(2n); a(n+1) = a(n) XOR 4*a(n) where XOR is binary exclusive OR operator. - Philippe Deléham, Jun 18 2005

a(n) = A001317(2n). - Alex Ratushnyak, May 04 2012

MAPLE

bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

# A recursive, cellular automaton rule version:

sigmaminus := proc(n) option remember: if (0 = n) then (1)

else sum('((bit_n(sigmaminus(n-1), i)+bit_n(sigmaminus(n-1), i-2)) mod 2)*(2^i)', 'i'=0..(2*n)) fi: end:

MATHEMATICA

r = 24; c = CellularAutomaton[90, {{1}, 0}, r - 1]; Table[FromDigits[c[[k, r - k + 1 ;; r + k - 1]], 2], {k, r}] (* Arkadiusz Wesolowski, Jun 09 2013 *)

PROG

(Python)

a=1

for n in range(55):

. print a,

. a ^= a*4

# Alex Ratushnyak, May 04 2012

CROSSREFS

Cf. A006977, A006978, A038184, A038185 (other cellular automata), A000215 (Fermat numbers).

Also alternate terms of A001317. Cf. A048710, A048720, A048757 (same 0/1-patterns interpreted in Fibonacci number system).

Equals 4*A089893(n)+1.

Sequence in context: A149754 A149755 A002020 * A149756 A036756 A149757

Adjacent sequences:  A038180 A038181 A038182 * A038184 A038185 A038186

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 09 1999

STATUS

approved

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Last modified April 24 02:19 EDT 2014. Contains 240947 sequences.