A265283 - OEIS

%I #29 Feb 16 2025 08:33:27

%S 1,3,4,6,6,8,8,10,10,12,12,14,14,16,16,18,18,20,20,22,22,24,24,26,26,

%T 28,28,30,30,32,32,34,34,36,36,38,38,40,40,42,42,44,44,46,46,48,48,50,

%U 50,52,52,54,54,56,56,58,58,60,60,62,62,64,64,66,66,68

%N Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

%C From _Gus Wiseman_, Apr 13 2019: (Start)

%C Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:

%C   (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)

%C         (31)   (41)    (42)     (52)      (53)       (63)        (64)

%C         (211)  (221)   (51)     (61)      (62)       (72)        (73)

%C                (2111)  (222)    (2221)    (71)       (81)        (82)

%C                        (2211)   (22111)   (2222)     (22221)     (91)

%C                        (21111)  (211111)  (22211)    (222111)    (22222)

%C                                           (221111)   (2211111)   (222211)

%C                                           (2111111)  (21111111)  (2221111)

%C                                                                  (22111111)

%C                                                                  (211111111)

%C (End)

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A265283/b265283.txt">Table of n, a(n) for n = 0..999</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000533">St000533: The maximal number of non-attacking rooks on a Ferrers shape</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

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

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Dec 07 2015 and Apr 16 2019: (Start)

%F a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.

%F a(n) = n+2 for n>1 and even.

%F a(n) = n+3 for n>1 and odd.

%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.

%F G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).

%F (End)

%e From _Michael De Vlieger_, Dec 14 2015: (Start)

%e First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:

%e                         1                          =  1

%e                       1 1 1                        =  3

%e                     1 1 . 1 1                      =  4

%e                   1 1 1 . 1 1 1                    =  6

%e                 1 1 . 1 . 1 . 1 1                  =  6

%e               1 1 1 . 1 . 1 . 1 1 1                =  8

%e             1 1 . 1 . 1 . 1 . 1 . 1 1              =  8

%e           1 1 1 . 1 . 1 . 1 . 1 . 1 1 1            = 10

%e         1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1          = 10

%e       1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1        = 12

%e     1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1      = 12

%e   1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1    = 14

%e 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1  = 14

%e (End)

%t rule = 94; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]

%t Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* _Michael De Vlieger_, Dec 14 2015 *)

%Y Column k = 2 of A325227.

%Y Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.

%K nonn,easy,changed

%O 0,2

%A _Robert Price_, Dec 06 2015