%I #48 Dec 16 2021 11:38:34

%S 1,7,21,107,273,1911,5189,28123,65793,460551,1381653,7039851,17829905,

%T 124809335,340873541,1840690907,4295032833,30065229831,90195689493,

%U 459568513131,1172543963409,8207807743863,22286925370437

%N 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 Generation n (starting from the generation 0: 1) interpreted as a binary number, but written in base 10.

%C Rows of the mod 2 trinomial triangle (A027907), interpreted as binary numbers: 1, 111, 10101, 1101011, ... (A118110). - _Jacob A. Siehler_, Aug 25 2006

%C See A071053 for number of ON cells. - _N. J. A. Sloane_, Jul 28 2014

%H Gheorghe Coserea, <a href="/A038184/b038184.txt">Table of n, a(n) for n = 0..200</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule150.html">Rule 150</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%e Bit patterns with "0" replaced by "." for visibilty [_Georg Fischer_, Dec 16 2021]:

%e 0: 1

%e 1: 111

%e 2: 1.1.1

%e 3: 11.1.11

%e 4: 1...1...1

%e 5: 111.111.111

%e 6: 1.1...1...1.1

%e 7: 11.11.111.11.11

%e 8: 1.......1.......1

%e 9: 111.....111.....111

%e 10: 1.1.1...1.1.1...1.1.1

%e 11: 11.1.11.11.1.11.11.1.11

%e 12: 1...1.......1.......1...1

%e 13: 111.111.....111.....111.111

%e 14: 1.1...1.1...1.1.1...1.1...1.1

%e 15: 11.11.11.11.11.1.11.11.11.11.11

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

%p sigmagen := proc(n) option remember: if (0 = n) then (1)

%p 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 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 (PARI)

%o a(n) = subst(lift(Pol(Mod([1,1,1],2),'x)^n),'x,2);

%o vector(23,n,a(n-1)) \\ _Gheorghe Coserea_, Jun 12 2016

%Y Cf. A006977, A006978, A038183, A038185 (other cellular automata).

%Y Cf. A048710, A048720, A027907, A001317, A071053.

%Y This sequence, A071036 and A118110 are equivalent descriptions of the Rule 150 automaton.

%K nonn

%O 0,2

%A _Antti Karttunen_, Feb 15 1999