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 1-5 of 5 results.

A290660 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 101, 1111, 11101, 111111, 1111101, 11111111, 111111101, 1111111111, 11111111101, 111111111111, 1111111111101, 11111111111111, 111111111111101, 1111111111111111, 11111111111111101, 111111111111111111, 1111111111111111101, 11111111111111111111
Offset: 0

Views

Author

Robert Price, Aug 08 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A166956, A290661, A290662.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Formula

Conjectures from Chai Wah Wu, Aug 03 2020: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 3.
G.f.: (100*x^3 - 10*x^2 + x + 1)/((x - 1)*(x + 1)*(10*x - 1)). (End)

A290661 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 101, 1111, 10111, 111111, 1011111, 11111111, 101111111, 1111111111, 10111111111, 111111111111, 1011111111111, 11111111111111, 101111111111111, 1111111111111111, 10111111111111111, 111111111111111111, 1011111111111111111, 11111111111111111111
Offset: 0

Views

Author

Robert Price, Aug 08 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A166956, A290660, A290662.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Formula

Conjectures from Chai Wah Wu, Aug 03 2020: (Start)
a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n > 3.
G.f.: (10*x^3 - 10*x^2 + 10*x + 1)/((x - 1)*(10*x - 1)*(10*x + 1)). (End)

A290662 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 15, 23, 63, 95, 255, 383, 1023, 1535, 4095, 6143, 16383, 24575, 65535, 98303, 262143, 393215, 1048575, 1572863, 4194303, 6291455, 16777215, 25165823, 67108863, 100663295, 268435455, 402653183, 1073741823, 1610612735, 4294967295, 6442450943
Offset: 0

Views

Author

Robert Price, Aug 08 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A166956, A290660, A290661.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 899; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Formula

Conjectures from Chai Wah Wu, Aug 03 2020: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 3.
G.f.: (2*x^3 - 2*x^2 + 2*x + 1)/((x - 1)*(2*x - 1)*(2*x + 1)). (End)

A166977 Jacobsthal-Lucas numbers A014551, except a(0) = 0.

Original entry on oeis.org

0, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647
Offset: 0

Views

Author

Paul Curtz, Oct 26 2009

Keywords

Comments

The sequence (-1)^n*a(n) is the inverse binomial transform of A166956.
The main diagonal of the table of a(n) and its higher differences in successive rows is 0,4,8,16,32,.. , 4*A131577(n).

Links

Programs

  • Mathematica
    Join[{0, 1}, LinearRecurrence[{1, 2}, {5, 7}, 50]] (* or *) Table[2^n + (-1)^n, {n,1,25}] (* G. C. Greubel, May 30 2016 *)

Formula

a(n) = A014551(n), n>0.
a(n) - A001045(n) = A097073(n), n>0.
a(n) - A001045(n) = 4*A001045(n-1).
a(n) = a(n-1) + 2*a(n-2), n>2.
G.f.: x*(1 + 4*x)/((1+x) * (1-2*x)).
a(n) = (-1)^n + 2^n for n>0. - Colin Barker, Jun 06 2012
E.g.f.: exp(2*x) + exp(-x) - 2. - G. C. Greubel, May 30 2016

Extensions

Edited and extended by R. J. Mathar, Mar 14 2010

A166978 a(n) = 4*( 1-(-1)^n) -2^n.

Original entry on oeis.org

-1, 6, -4, 0, -16, -24, -64, -120, -256, -504, -1024, -2040, -4096, -8184, -16384, -32760, -65536, -131064, -262144, -524280, -1048576, -2097144, -4194304, -8388600, -16777216, -33554424, -67108864, -134217720, -268435456, -536870904, -1073741824, -2147483640
Offset: 0

Views

Author

Paul Curtz, Oct 26 2009

Keywords

Links

Programs

  • Magma
    [4*( 1-(-1)^n) -2^n: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011
  • Mathematica
    LinearRecurrence[{2,1,-2}, {-1, 6, -4}, 50] (* or *) Table[4*(1-(-1)^n) - 2^n, {n,0,25}] (* G. C. Greubel, May 30 2016 *)

Formula

a(n) = A166956(n+1)-3*A166956(n).
a(2n) = -A000302(n). a(2n+1) = 6*(-1)^n*A084240(n).
a(n+1) - 2*a(n) = 4*( 3*(-1)^n-1) = 8 *(-1)^n*A000034(n).
G.f.: -(5*x-1)*(3*x-1) / ( (x-1)*(2*x-1)*(1+x) ). - R. J. Mathar, Jul 01 2011
E.g.f.: 8*sinh(x) - exp(2*x). - G. C. Greubel, May 30 2016
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.