cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A267705 Binary representation of the n-th iteration of the "Rule 205" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 10, 10101, 1101011, 111010111, 11110101111, 1111101011111, 111111010111111, 11111110101111111, 1111111101011111111, 111111111010111111111, 11111111110101111111111, 1111111111101011111111111, 111111111111010111111111111, 11111111111110101111111111111
Offset: 0

Views

Author

Robert Price, Jan 19 2016

Keywords

References

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

Links

Crossrefs

Cf. A267704, A188530.

Programs

  • Mathematica
    rule=205; 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]]],{k,1,rows}]   (* Binary Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 20 2016 and Apr 20 2019: (Start)
a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3) for n>3.
G.f.: (1-101*x+10101*x^2-10100*x^3) / ((1-x)*(1-10*x)*(1-100*x)).
(End)

A266176 Decimal representation of the n-th iteration of the "Rule 5" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 4, 107, 16, 1967, 64, 32447, 256, 523007, 1024, 8383487, 4096, 134197247, 16384, 2147401727, 65536, 34359410687, 262144, 549754503167, 1048576, 8796087779327, 4194304, 140737467383807, 16777216, 2251799729799167, 67108864, 36028796683419647, 268435456
Offset: 0

Views

Author

Robert Price, Dec 22 2015

Keywords

References

  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A266174, A266175, A000079, A188530.

Programs

  • Mathematica
    rule = 5; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]
  • Python
    print([2*4**n -5*2**(n-1) -1 if n%2 else 2**n for n in range(30)]) # Karl V. Keller, Jr., Jun 20 2021

Formula

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)
a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.
G.f.: (1+2*x-17*x^2+65*x^3+16*x^4-112*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).
(End)
a(n) = 2^n = A000079(n) for even n>=0; a(n) = 2*4^n - 5*2^(n-1) - 1 = A188530(n) for odd n. - Karl V. Keller, Jr., Jun 19 2021
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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