A266380 Decimal representation of the n-th iteration of the "Rule 21" elementary cellular automaton starting with a single ON (black) cell.
1, 3, 0, 127, 0, 2047, 0, 32767, 0, 524287, 0, 8388607, 0, 134217727, 0, 2147483647, 0, 34359738367, 0, 549755813887, 0, 8796093022207, 0, 140737488355327, 0, 2251799813685247, 0, 36028797018963967, 0, 576460752303423487, 0, 9223372036854775807, 0
Offset: 0
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Links
- Robert Price, Table of n, a(n) for n = 0..500
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).
Crossrefs
Programs
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Magma
[n le 1 select 3^n else (1-(-1)^n)*(4*16^Floor(n/2)-1/2): n in [0..40]]; // Bruno Berselli, Dec 29 2015
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Mathematica
rule=21; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}] (* Decimal Representation of Rows *) -
Python
print([(2*4**n - 1)*(n%2) + 0**n - 4*0**abs(n-1) for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021
Formula
From Colin Barker, Dec 29 2015 and Apr 15 2019: (Start)
a(n) = 17*a(n-2) - 16*a(n-4) for n>5.
G.f.: (1 + 3*x - 17*x^2 + 76*x^3 + 16*x^4 - 64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)). (End)
a(n) = (1 - (-1)^n)*(4*16^floor(n/2) - 1/2) for n>1. - Bruno Berselli, Dec 29 2015
a(n) = (2*4^n - 1)*(n mod 2) + 0^n - 4*0^abs(n-1). - Karl V. Keller, Jr., Sep 03 2021
Comments