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.

A284302 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 865", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 6, 10, 10, 46, 42, 238, 426, 750, 682, 2798, 2730, 10990, 10922, 44014, 43946, 178926, 174762, 781038, 764586, 3844846, 7252650, 11185134, 16427946, 48937710, 45787818, 178973422, 178956970, 715827950, 715827882, 2863311854, 2863311786, 11453250286
Offset: 0

Views

Author

Robert Price, Mar 24 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. A284299, A284300, A284301.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 865; 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 Chai Wah Wu, May 04 2024: (Start)
a(n) = 4*a(n-2) + a(n-8) - 4*a(n-10) for n > 36.
G.f.: (4194304*x^36 + 16777216*x^35 + 19922944*x^34 - 4194304*x^33 + 12582912*x^32 + 4194304*x^31 - 4194304*x^30 - 786432*x^29 - 4259840*x^28 - 16842752*x^27 - 19922944*x^26 + 4194304*x^25 - 12582912*x^24 - 4194304*x^23 + 4194304*x^22 + 720896*x^21 + 65536*x^20 + 65536*x^19 + 3072*x^17 - 208*x^13 + 16*x^12 - 212*x^11 - 1024*x^10 - 202*x^9 + 257*x^8 + 54*x^7 + 2*x^6 + 6*x^5 - 14*x^4 + 10*x^3 + 2*x^2 + 1)/(4*x^10 - x^8 - 4*x^2 + 1). (End)

A284299 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 865", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 11, 101, 1010, 11101, 101010, 1110111, 10101011, 111011101, 1010101010, 11101110101, 101010101010, 1110111010101, 10101010101010, 111011111010101, 1010101110101010, 11101110101110101, 101010101010101010, 1110111010101111101, 10101010101010111010
Offset: 0

Views

Author

Robert Price, Mar 24 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. A284300, A284301, A284302.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 865; 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 Chai Wah Wu, May 04 2024: (Start)
a(n) = a(n-2) + 100000000*a(n-8) - 100000000*a(n-10) for n > 36.
G.f.: (100000000000000*x^36 + 100000000000*x^35 - 98990000000000*x^34 - 100000000000*x^33 - 9900000000*x^32 + 1000000000*x^31 - 100000000*x^30 + 99000000000*x^29 - 1000001000000*x^28 - 100000001000*x^27 + 989900*x^26 + 1000*x^25 + 99*x^24 - 10*x^23 + x^22 - 100990*x^21 + 10000*x^20 + 1000*x^19 - 9900000*x^17 - 990100000*x^13 + 100000000*x^12 + 890099000*x^11 - x^10 + 109900990*x^9 - 89999999*x^8 + 1099010*x^7 + 100000*x^6 + 11000*x^5 + 999*x^4 + 101*x^3 + 10*x^2 + 1)/(100000000*x^10 - 100000000*x^8 - x^2 + 1). (End)

A284300 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 865", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 110, 1010, 1010, 101110, 101010, 11101110, 110101010, 1011101110, 1010101010, 101011101110, 101010101010, 10101011101110, 10101010101010, 1010101111101110, 1010101110101010, 101011101011101110, 101010101010101010, 10111110101011101110
Offset: 0

Views

Author

Robert Price, Mar 24 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. A284299, A284301, A284302.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 865; 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, May 04 2024: (Start)
a(n) = 100*a(n-2) + a(n-8) - 100*a(n-10) for n > 36.
G.f.: (10000000000000000000000*x^36 + 1000000000000000000000000*x^35 + 1009900000000000000000000*x^34 - 10000000000000000000000*x^33 + 990000000000000000000000*x^32 + 10000000000000000000000*x^31 - 10000000000000000000000*x^30 - 99000000000000000000*x^29 - 10000010000000000000000*x^28 - 1000000010000000000000000*x^27 - 1009900000000000000000000*x^26 + 10000000000000000000000*x^25 - 990000000000000000000000*x^24 - 10000000000000000000000*x^23 + 10000000000000000000000*x^22 + 98990000000000000000*x^21 + 10000000000000000*x^20 + 10000000000000000*x^19 + 990000000000*x^17 - 99010000*x^13 + 10000*x^12 - 99010900*x^11 - 10000000000*x^10 - 99009890*x^9 + 100000009*x^8 + 990110*x^7 + 10*x^6 + 110*x^5 - 9990*x^4 + 1010*x^3 + 10*x^2 + 1)/(100*x^10 - x^8 - 100*x^2 + 1). (End)

A285302 Positions of 0 in A285301, complement of A086398.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98, 100, 101, 102
Offset: 1

Views

Author

Clark Kimberling, Apr 25 2017

Keywords

Comments

Conjecture: -2 < n*r - a(n) < 2 for n>=1, where r = 1 + sqrt(1/3).

Examples

			As a word, A284301 = 100010..., in which the positions of 0 are 2,3,4,6,8,...

Links

Crossrefs

Cf. A284301, A086398.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0, 0}}] &, {0}, 10]; (* A285301 *)
u = Flatten[Position[s, 0]];  (* A285302 *)
v = Flatten[Position[s, 1]];  (* A086398 *)
Showing 1-4 of 4 results.

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