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A038184 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, converted to a decimal number.

%I A038184 #49 Feb 16 2025 08:32:38
%S A038184 1,7,21,107,273,1911,5189,28123,65793,460551,1381653,7039851,17829905,
%T A038184 124809335,340873541,1840690907,4295032833,30065229831,90195689493,
%U A038184 459568513131,1172543963409,8207807743863,22286925370437
%N A038184 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, converted to a decimal number.
%C A038184 Generation n (starting from the generation 0: 1) interpreted as a binary number, but written in base 10.
%C A038184 Rows of the mod 2 trinomial triangle (A027907), interpreted as binary numbers: 1, 111, 10101, 1101011, ... (A118110). - _Jacob A. Siehler_, Aug 25 2006
%C A038184 See A071053 for number of ON cells. - _N. J. A. Sloane_, Jul 28 2014
%H A038184 Gheorghe Coserea, <a href="/A038184/b038184.txt">Table of n, a(n) for n = 0..200</a>
%H A038184 Alan J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/CellularAutomatonRule150.ps">On generating functions of some sequences of integers defined in the evolution of the cellular automaton Rule 150</a>, Preprint 2016.
%H A038184 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule150.html">Rule 150</a>
%H A038184 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A038184 Bit patterns with "0" replaced by "." for visibilty [_Georg Fischer_, Dec 16 2021]:
%e A038184   0:                    1
%e A038184   1:                   111
%e A038184   2:                  1.1.1
%e A038184   3:                 11.1.11
%e A038184   4:                1...1...1
%e A038184   5:               111.111.111
%e A038184   6:              1.1...1...1.1
%e A038184   7:             11.11.111.11.11
%e A038184   8:            1.......1.......1
%e A038184   9:           111.....111.....111
%e A038184   10:         1.1.1...1.1.1...1.1.1
%e A038184   11:        11.1.11.11.1.11.11.1.11
%e A038184   12:       1...1.......1.......1...1
%e A038184   13:      111.111.....111.....111.111
%e A038184   14:     1.1...1.1...1.1.1...1.1...1.1
%e A038184   15:    11.11.11.11.11.1.11.11.11.11.11
%p A038184 bit_n := (x,n) -> `mod`(floor(x/(2^n)),2);
%p A038184 sigmagen := proc(n) option remember: if (0 = n) then (1)
%p A038184 else sum('((bit_n(sigmagen(n-1),i)+bit_n(sigmagen(n-1),i-1)+bit_n(sigmagen(n-1),i-2)) mod 2)*(2^i)', 'i'=0..(2*n)) fi: end:
%t A038184 f[n_] := Sum[2^k*Coefficient[ #, x, k], {k, 0, 2n}] & @ Expand[(1 + x + x^2)^n, Modulus -> 2] (* _Jacob A. Siehler_, Aug 25 2006 *)
%o A038184 (PARI)
%o A038184 a(n) = subst(lift(Pol(Mod([1,1,1],2),'x)^n),'x,2);
%o A038184 vector(23,n,a(n-1))  \\ _Gheorghe Coserea_, Jun 12 2016
%Y A038184 Cf. A006977, A006978, A038183, A038185 (other cellular automata).
%Y A038184 Cf. A048710, A048720, A027907, A001317, A071053.
%Y A038184 This sequence, A071036 and A118110 are equivalent descriptions of the Rule 150 automaton.
%K A038184 nonn
%O A038184 0,2
%A A038184 _Antti Karttunen_, Feb 15 1999