OFFSET
0,2
COMMENTS
a(1660) is 1000 digits long. - Michael De Vlieger, Oct 07 2015
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1660
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig 8.
Eric Weisstein's World of Mathematics, Rule 54
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).
FORMULA
a(n) = 7*(4^(n+1)-1)/15 for n odd; a(n) = (4^(n+2)-1)/15 for n even.
From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)
a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
G.f.: (7*x+1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)
a(n) = floor((16+12*(n mod 2))*4^n/15). - Karl V. Keller, Jr., Aug 04 2021
EXAMPLE
From Michael De Vlieger, Oct 07 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right:
1 = 1
111 = 7
1 0001 = 17
111 0111 = 119
1 0001 0001 = 273
111 0111 0111 = 1911
1 0001 0001 0001 = 4369
111 0111 0111 0111 = 30583
1 0001 0001 0001 0001 = 69905
(End)
MATHEMATICA
clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[54, {{1}, 0}, 27]] (* or *)
Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* Michael De Vlieger, Oct 07 2015 *)
PROG
(Python) print([(16+12*(n%2))*4**n//15 for n in range(30)]) # Karl V. Keller, Jr., Aug 04 2021
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Eric W. Weisstein, Apr 13 2006
EXTENSIONS
a(23)-a(24) from Michael De Vlieger, Oct 07 2015
STATUS
approved