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.

A266255 Decimal representation of the n-th iteration of the "Rule 11" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 3, 124, 3, 2044, 3, 32764, 3, 524284, 3, 8388604, 3, 134217724, 3, 2147483644, 3, 34359738364, 3, 549755813884, 3, 8796093022204, 3, 140737488355324, 3, 2251799813685244, 3, 36028797018963964, 3, 576460752303423484, 3, 9223372036854775804, 3
Offset: 0

Views

Author

Robert Price, Dec 25 2015

Keywords

Comments

Rule 43 also generates this sequence.

References

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

Links

Crossrefs

Cf. A266253, A266254, A266070, A266071, A081253, A266256, A266257, A266258, A266259.

Programs

  • Mathematica
    rule=11; 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 - 4 if n%2 else 3 - 2*0**n for n in range(33)]) # Karl V. Keller, Jr., Aug 26 2021

Formula

From Colin Barker, Dec 27 2015 and Apr 14 2019: (Start)
a(n) = (7*(-1)^n+2^(2*n+1)-(-1)^n*2^(2*n+1)-1)/2 for n>0.
a(n) = 17*a(n-2)-16*a(n-4) for n>4.
G.f.: (1+4*x-14*x^2+56*x^3-32*x^4) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).
(End)
a(n) = 2*4^n - 4 for odd n; a(n) = 3 - 2*0^n for even n. - Karl V. Keller, Jr., Aug 26 2021

A266662 Number of ON (black) cells in the n-th iteration of the "Rule 47" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 2, 5, 2, 9, 2, 13, 2, 17, 2, 21, 2, 25, 2, 29, 2, 33, 2, 37, 2, 41, 2, 45, 2, 49, 2, 53, 2, 57, 2, 61, 2, 65, 2, 69, 2, 73, 2, 77, 2, 81, 2, 85, 2, 89, 2, 93, 2, 97, 2, 101, 2, 105, 2, 109, 2, 113, 2, 117, 2, 121, 2, 125, 2, 129, 2, 133, 2, 137, 2, 141
Offset: 0

Views

Author

Robert Price, Jan 02 2016

Keywords

References

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

Links

Crossrefs

Cf. A266659.

Programs

  • Mathematica
    rule=47; 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[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

Formula

Conjectures from Colin Barker, Jan 03 2016 and Apr 18 2019: (Start)
a(n) = (-2*(-1)^n*n+2*n+3*(-1)^n+1)/2 for n>1.
a(n) = 2*a(n-2)-a(n-4) for n>5.
G.f.: (1+2*x+x^3-x^4+x^5) / ((1-x)^2*(1+x)^2).
(End)
a(n) = A266256(n), n>1. - R. J. Mathar, Jan 10 2016
Showing 1-2 of 2 results.

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

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