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.

A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

Original entry on oeis.org

4, 24, 112, 480, 1984, 8064, 32512, 130560, 523264, 2095104, 8384512, 33546240, 134201344, 536838144, 2147418112, 8589803520, 34359476224, 137438429184, 549754765312, 2199021158400, 8796088827904, 35184363700224, 140737471578112, 562949919866880
Offset: 0

Views

Author

Jonas Wallgren, Feb 02 2001

Keywords

Comments

A hierarchical sequence (S(W'2{2}c) - see A059126).
a(n) written in base 2: 100, 11000, 1110000, ..., i.e., (n+1) times 1 and (n+2) times 0 (see A163663). - Jaroslav Krizek, Aug 12 2009
Also, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood. - Robert Price, May 04 2016

References

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

Links

Crossrefs

Cf. A006516, A020522, A059126, A163663, A173787, A272702.

Programs

  • Mathematica
    Table[2^(n + 2)*(2^(n + 1) - 1), {n, 0, 23}] (* and *) LinearRecurrence[{6, -8}, {4, 24}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
  • PARI
    a(n) = { 2^(n + 2)*(2^(n + 1) - 1) } \\ Harry J. Smith, Jun 25 2009

Formula

a(n) = A173787(2*n+3,n+2) = 4*A006516(n+1). - Reinhard Zumkeller, Feb 28 2010
From Colin Barker, Apr 28 2013: (Start)
a(n) = 6*a(n-1) - 8*a(n-2).
G.f.: 4 / ((2*x-1)*(4*x-1)). (End)
a(n) = 2*A020522(n+1). - Hussam al-Homsi, Jun 06 2021
E.g.f.: 4*exp(2*x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Dec 10 2023

Extensions

Revised by Henry Bottomley, Jun 27 2005

A272703 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 18, 42, 95, 160, 273, 385, 614, 855, 1176, 1465, 1970, 2459, 3060, 3540, 4505, 5482, 6603, 7628, 9061, 10414, 11943, 13160, 15313, 17418, 19731, 21764, 24517, 27006, 29735, 31719, 35692, 39677, 43934, 47967, 52792, 57409, 62330, 66555, 72612, 78493
Offset: 0

Views

Author

Robert Price, May 04 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. A272702.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=513; 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 *)

A272704 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

3, 9, 11, 29, 12, 48, -1, 117, 12, 80, -32, 216, -16, 112, -121, 485, 12, 144, -96, 408, -80, 176, -312, 936, -48, 208, -280, 720, -264, 240, -745, 1989, 12, 272, -224, 792, -208, 304, -696, 1832, -176, 336, -664, 1360, -648, 368, -1640, 3912, -112, 400
Offset: 0

Views

Author

Robert Price, May 04 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. A272702.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=513; 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 *)
Showing 1-3 of 3 results.

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