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 1-3 of 3 results.

A273419 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 26, 67, 139, 252, 412, 629, 909, 1262, 1694, 2215, 2831, 3552, 4384, 5337, 6417, 7634, 8994, 10507, 12179, 14020, 16036, 18237, 20629, 23222, 26022, 29039, 32279, 35752, 39464, 43425, 47641, 52122, 56874, 61907, 67227, 72844, 78764, 84997, 91549, 98430
Offset: 0

Views

Author

Robert Price, May 22 2016

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. A273417.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=705; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)

Formula

Conjectures from Colin Barker, May 22 2016: (Start)
a(n) = (111-3*(-1)^n-58*n+48*n^2+16*n^3)/12 for n>1.
a(n) = (8*n^3+24*n^2-29*n+54)/6 for n>1 and even.
a(n) = (8*n^3+24*n^2-29*n+57)/6 for n>1 and odd.
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>6.
G.f.: (1+2*x+13*x^2+x^3-3*x^4+7*x^5-5*x^6) / ((1-x)^4*(1+x)).
(End)

A273420 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

3, 17, 20, 31, 41, 47, 57, 63, 73, 79, 89, 95, 105, 111, 121, 127, 137, 143, 153, 159, 169, 175, 185, 191, 201, 207, 217, 223, 233, 239, 249, 255, 265, 271, 281, 287, 297, 303, 313, 319, 329, 335, 345, 351, 361, 367, 377, 383, 393, 399, 409, 415, 425, 431
Offset: 0

Views

Author

Robert Price, May 22 2016

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. A273417.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=705; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)

Formula

Conjectures from Colin Barker, May 22 2016: (Start)
a(n) = 8+(-1)^n+8*n for n>2.
a(n) = 8*n+9 for n>2 and even.
a(n) = 8*n+7 for n>2 and odd.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>5.
G.f.: (3+14*x-3*x^3+7*x^4-5*x^5) / ((1-x)^2*(1+x)).
(End)

A273418 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 41, 217, 953, 3961, 16121, 65017, 261113, 1046521, 4190201, 16769017, 67092473, 268402681, 1073676281, 4294836217
Offset: 0

Views

Author

Robert Price, May 22 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Conjecture: Rule 713 also generates this sequence. - Lars Blomberg, Jul 19 2016

References

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

Links

Crossrefs

Cf. A273417.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=705; 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}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Formula

Conjecture: a(n) = A273313(n), n>1. - R. J. Mathar, May 27 2016
Conjecture: a(n) = 4*4^n - 4*2^n - 7, n>1. - Lars Blomberg, Jul 19 2016
Conjectures from Colin Barker, Dec 01 2016: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>4.
G.f.: (1 - 3*x + 27*x^2 - 22*x^3 - 24*x^4) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

Extensions

a(8)-a(15) from Lars Blomberg, Jul 19 2016
Showing 1-3 of 3 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.