OFFSET
0,2
COMMENTS
From Gus Wiseman, Apr 13 2019: (Start)
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:
(21) (22) (32) (33) (43) (44) (54) (55)
(31) (41) (42) (52) (53) (63) (64)
(211) (221) (51) (61) (62) (72) (73)
(2111) (222) (2221) (71) (81) (82)
(2211) (22111) (2222) (22221) (91)
(21111) (211111) (22211) (222111) (22222)
(221111) (2211111) (222211)
(2111111) (21111111) (2221111)
(22111111)
(211111111)
(End)
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
FORMULA
Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.
a(n) = n+2 for n>1 and even.
a(n) = n+3 for n>1 and odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).
(End)
EXAMPLE
From Michael De Vlieger, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
1 = 1
1 1 1 = 3
1 1 . 1 1 = 4
1 1 1 . 1 1 1 = 6
1 1 . 1 . 1 . 1 1 = 6
1 1 1 . 1 . 1 . 1 1 1 = 8
1 1 . 1 . 1 . 1 . 1 . 1 1 = 8
1 1 1 . 1 . 1 . 1 . 1 . 1 1 1 = 10
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 10
1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1 = 12
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 12
1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1 = 14
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 = 14
(End)
MATHEMATICA
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}]
Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* Michael De Vlieger, Dec 14 2015 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Dec 06 2015
STATUS
approved