A265381 Decimal representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.
1, 3, 7, 14, 29, 59, 119, 238, 477, 955, 1911, 3822, 7645, 15291, 30583, 61166, 122333, 244667, 489335, 978670, 1957341, 3914683, 7829367, 15658734, 31317469, 62634939, 125269879, 250539758, 501079517, 1002159035, 2004318071, 4008636142, 8017272285
Offset: 0
Examples
From _Michael De Vlieger_, Dec 09 2015: (Start)
First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses. The center column values up to that row are concatenated then converted into decimal at right:
Rule 158 Binary Decimal
(1) -> 1 = 1
1 (1) 1 -> 11 = 3
1 1 (1) 0 1 -> 111 = 7
1 1 1 (0) 0 1 1 -> 1110 = 14
1 1 1 0 (1) 1 1 0 1 -> 11101 = 29
1 1 1 0 0 (1) 1 0 0 1 1 -> 111011 = 59
1 1 1 0 1 1 (1) 0 1 1 1 0 1 -> 1110111 = 119
1 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1 -> 11101110 = 238
1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1 -> 111011101 = 477
(End)
Links
- Robert Price, Table of n, a(n) for n = 0..999
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Eric Weisstein's World of Mathematics, Rule 158
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,-2).
Programs
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Mathematica
f[n_] := Block[{w = {}}, Do[AppendTo[w, Boole[Mod[k, 4] != 3]], {k, 0, n}]; FromDigits[w, 2]]; Table[f@ n, {n, 0, 32}] (* Michael De Vlieger, Dec 09 2015 *) -
Python
print([7*2**(n+2)//15 for n in range(34)]) # Karl V. Keller, Jr., Oct 01 2020
Formula
From Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (-45+5*(-1)^n-(6-i*3)*(-i)^n-(6+3*i)*i^n+7*2^(4+n))/60 where i = sqrt(-1).
a(n) = 2*a(n-1)+a(n-4)-2*a(n-5) for n>4.
G.f.: (1+x+x^2) / ((1-x)*(1+x)*(1-2*x)*(1+x^2)).
(End)
a(n) = floor(7*2^(n+2)/15) for n>=0. - Karl V. Keller, Jr., Oct 01 2020