cp's OEIS Frontend

A327982 Partial sums of A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

%I A327982 #18 May 31 2023 09:11:43
%S A327982 1,2,2,3,4,5,5,5,6,7,7,7,7,8,8,9,10,10,10,11,11,11,12,13,14,14,15,15,
%T A327982 16,17,18,18,18,19,20,21,21,22,22,23,23,24,25,25,25,25,25,26,27,27,27,
%U A327982 28,28,29,29,30,31,31,32,32,33,33,34,35,36,37,38,39,39,39,39,39,40,41,42,43,43,43,43,44,44,45,45,46
%N A327982 Partial sums of A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.
%C A327982 Lexicographically earliest monotonic left inverse of A327984.
%C A327982 Proving (or disproving) that Lim_{n->inf} a(n)/n = 1/2 would solve the Problem 2: "Does each color of cell occur on average equally often in the center column?" of Stephen Wolfram's 2019 prize announcement.
%H A327982 Antti Karttunen, <a href="/A327982/b327982.txt">Table of n, a(n) for n = 0..100000</a>
%H A327982 Stephen Wolfram, <a href="https://blog.wolfram.com/2019/10/01/announcing-the-rule-30-prizes/">Announcing the Rule 30 Prizes</a>, 2019
%H A327982 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A327982 a(0) = A051023(0) = 1, for n > 0, a(n) = A051023(n) + a(n-1).
%F A327982 For all n >= 0, a(A327984(n)) = n.
%e A327982 The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
%e A327982 ---------------------------------------------- a(n)
%e A327982    0:              (1)                          1
%e A327982    1:             1(1)1                         2
%e A327982    2:            11(0)01                        2
%e A327982    3:           110(1)111                       3
%e A327982    4:          1100(1)0001                      4
%e A327982    5:         11011(1)10111                     5
%e A327982    6:        110010(0)001001                    5
%e A327982    7:       1101111(0)0111111                   5
%e A327982    8:      11001000(1)11000001                  6
%e A327982    9:     110111101(1)001000111                 7
%e A327982   10:    1100100001(0)1111011001                7
%e A327982   11:   11011110011(0)10000101111               7
%e A327982   12:  110010001110(0)110011010001              7
%e A327982   13: 1101111011001(1)1011100110111             8
%e A327982 We count how many 1's have occurred so far in the central column (indicated with parentheses), giving us the terms: 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 7, 7, 8, ....
%t A327982 A327982list[nmax_]:=Accumulate[CellularAutomaton[30,{{1},0},{nmax,{{0}}}]];A327982list[100] (* _Paolo Xausa_, May 30 2023 *)
%o A327982 (PARI)
%o A327982 A269160(n) = bitxor(n, bitor(2*n, 4*n)); \\ From A269160.
%o A327982 A110240(n) = if(!n,1,A269160(A110240(n-1)));
%o A327982 A051023(n) = ((A110240(n)>>n)%2);
%o A327982 A327982(n) = (A051023(n)+if(0==n,0,A327982(n-1)));
%o A327982 (PARI)
%o A327982 up_to = 105;
%o A327982 A269160(n) = bitxor(n, bitor(2*n, 4*n));
%o A327982 A327982list(up_to) = { my(v=vector(1+up_to), s=1, n=0, c=0); while(n<=up_to, c += (s>>n)%2; n++; v[n] = c; s = A269160(s)); (v); }
%o A327982 v327982 = A327982list(up_to);
%o A327982 A327982(n) = v327982[1+n];
%Y A327982 Cf. A051023, A110240, A269160, A327973, A327980, A327981, A327983, A327984.
%K A327982 nonn
%O A327982 0,2
%A A327982 _Antti Karttunen_, Oct 03 2019