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.

A282088 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 553", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 1, 0, 101, 0, 10101, 0, 1010101, 0, 101010101, 0, 10101010101, 0, 1010101010101, 0, 101010101010101, 0, 10101010101010101, 0, 1010101010101010101, 0, 101010101010101010101, 0, 10101010101010101010101, 0, 1010101010101010101010101, 0
Offset: 0

Views

Author

Robert Price, Feb 27 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. A282142, A282577, A282579.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 553; 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}]

Formula

Conjectures from Colin Barker, Feb 27 2017: (Start)
a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
a(n) = (10^n - 1) / 99 for n>0 and even.
a(n) = 0 for n odd.
G.f.: (1 - 100*x^2 + 100*x^4) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
(End)

A282142 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 553", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 100, 0, 10100, 0, 1010100, 0, 101010100, 0, 10101010100, 0, 1010101010100, 0, 101010101010100, 0, 10101010101010100, 0, 1010101010101010100, 0, 101010101010101010100, 0, 10101010101010101010100, 0, 1010101010101010101010100, 0
Offset: 0

Views

Author

Robert Price, Feb 27 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. A282088, A282577, A282579.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 553; 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 Colin Barker, Feb 27 2017: (Start)
a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
a(n) = 100*(10^n - 1) / 99 for n>0 and even.
a(n) = 0 for n odd.
G.f.: (1 - x^2 + 100*x^4) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
(End)

A282579 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 553", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 4, 0, 20, 0, 84, 0, 340, 0, 1364, 0, 5460, 0, 21844, 0, 87380, 0, 349524, 0, 1398100, 0, 5592404, 0, 22369620, 0, 89478484, 0, 357913940, 0, 1431655764, 0, 5726623060, 0, 22906492244, 0, 91625968980, 0, 366503875924, 0, 1466015503700, 0, 5864062014804
Offset: 0

Views

Author

Robert Price, Feb 27 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. A282088, A282142, A282577.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 553; 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}]

Formula

Conjectures from Colin Barker, Feb 27 2017: (Start)
a(n) = 4*(2^n - 1) / 3 for n>1 and even.
a(n) = 0 for n odd.
a(n) = 5*a(n-2) - 4*a(n-4) for n>4.
G.f.: (1 - x^2 + 4*x^4) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)

A366987 Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.

Original entry on oeis.org

-1, 0, 0, -2, -1, -2, 2, 1, -1, -2, -6, -3, -3, -3, -6, 10, 5, 1, -1, -5, -10, -22, -11, -7, -5, -7, -11, -22, 42, 21, 9, 3, -3, -9, -21, -42, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342
Offset: 0

Views

Author

Paul Curtz and Thomas Scheuerle, Oct 31 2023

Keywords

Examples

			Triangle T(n, k) starts:
   -1
    0   0
   -2  -1  -2
    2   1  -1  -2
   -6  -3  -3  -3  -6
   10   5   1  -1  -5 -10
  -22 -11  -7  -5  -7 -11 -22
   42  21   9   3  -3  -9 -21 -42
   ...
Note the symmetrical distribution of the absolute values of the terms in each row.

Links

Crossrefs

Rows sums: -A282577(n+2), if the conjectures from Colin Barker in A282577 are true.
First column: -(-1)^n * A078008(n).
Second column: (-1)^n * A001045(n).
Third column: -A140966(n).
Fourth column: (-1)^n * A155980(n+2).
Cf. A000079, A020989, A048573, A053088, A059570, A077898, A140966.

Programs

  • Maple
    T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Nov 02 2023
  • Mathematica
    A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3,{k,0,n}];Array[A366987row,15,0] (* Paolo Xausa, Nov 07 2023 *)
  • PARI
    T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023

Formula

T(n, 0) = -((-2)^n + 2)/3.
T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.
T(2*n+1, n) = A001045(n).
T(2*n+1, n+1) = -A001045(n).
T(2*n, n+1) = -A048573(n-1), for n > 0.
Note that the definition of T extends to negative parameters:
T(2*n-2, n-1) = -A001045(n).
-2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).
-4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).
-Sum_{k=0..n} (-1)^k*T(n, k) = A077898(n). See also A053088.
Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.
Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.
G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - Stefano Spezia, Nov 03 2023

Extensions

a(42) corrected by Paolo Xausa, Nov 07 2023
Showing 1-4 of 4 results.

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