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 1-4 of 4 results.

A062878 a(n) is the position of A050614(n) in A062877.

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 102, 255, 384, 960, 1632, 4080, 6168, 15420, 26214, 65535, 98304, 245760, 417792, 1044480, 1579008, 3947520, 6710784, 16776960, 25166208, 62915520, 106956384, 267390960, 404232216, 1010580540, 1717986918, 4294967295, 6442450944
Offset: 0

Views

Author

Antti Karttunen, Jun 26 2001

Keywords

Comments

In binary this sequence looks like 1, 11, 110, 1111, 11000, 111100, 1100110, 11111111, 110000000, 1111000000, 11001100000, 111111110000, 1100000011000, 11110000111100, 110011001100110, ...
Sequence A282387 may be the same, but I cannot prove nor disprove this beyond a(22). - Robert Price, Feb 13 2017
Agrees with A282387 for at least 1000 terms. - Sean A. Irvine, Apr 14 2023

Programs

  • Mathematica
    A050614 = Table[k = Floor[Log[2, n + 1]]; Product[j = 2^(i + 1); l = Fibonacci[j + 1] + Fibonacci[j - 1]; If[BitAnd[2^i, n] == 0, b = 0, b = 1]; l^b, {i, 0, k}], {n, 0, 200}]; A062877 = Union[Total /@ Subsets[Fibonacci[Range[1, 46, 2]]]]; Flatten[Table[Position[ A062877, A050614[[i]] ] - 1, {i, 1, 25}]] (* Robert Price, Feb 13 2017 *)

Formula

a(2^n-1) = 2^(2^n) - 1. - Philippe Deléham, Apr 05 2007
a(n) = Sum_{k=0..n} A127872(n,k)*2^k. - Philippe Deléham, Oct 09 2007

Extensions

a(15)-a(22) from Robert Price, Feb 13 2017
More terms from Sean A. Irvine, Apr 14 2023

A282385 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 462", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 110, 1111, 11000, 111100, 1100110, 11111111, 110000000, 1111000000, 11001100000, 111111110000, 1100000011000, 11110000111100, 110011001100110, 1111111111111111, 11000000000000000, 111100000000000000, 1100110000000000000, 11111111000000000000
Offset: 0

Views

Author

Robert Price, Feb 13 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. A282386, A282387, A282388.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 462; 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[1, i]], 10], {i, 1, stages - 1}]

A282386 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 462", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 11, 1111, 11, 1111, 110011, 11111111, 11, 1111, 110011, 11111111, 1100000011, 111100001111, 11001100110011, 1111111111111111, 11, 1111, 110011, 11111111, 1100000011, 111100001111, 11001100110011, 1111111111111111, 110000000000000011
Offset: 0

Views

Author

Robert Price, Feb 13 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. A282385, A282387, A282388.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 462; 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}]

A282388 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 462", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 15, 3, 15, 51, 255, 3, 15, 51, 255, 771, 3855, 13107, 65535, 3, 15, 51, 255, 771, 3855, 13107, 65535, 196611, 983055, 3342387, 16711935, 50529027, 252645135, 858993459, 4294967295, 3, 15, 51, 255, 771, 3855, 13107, 65535, 196611, 983055, 3342387
Offset: 0

Views

Author

Robert Price, Feb 13 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. A282385, A282386, A282387.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 462; 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]], 2], {i ,1, stages - 1}]
Showing 1-4 of 4 results.

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