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 1-2 of 2 results.

A255603 Partial sums of A253090.

Original entry on oeis.org

1, 10, 18, 50, 66, 154, 170, 274, 290, 426, 498, 810, 866, 1226, 1302, 1770, 1846, 2290, 2446, 3154, 3306, 4234, 4398, 5394, 5550, 6474, 6766, 8082, 8338, 9858, 10206, 11882, 12186, 13954, 14354, 16282, 16690, 18730, 19254, 21882, 22398, 25082, 25770, 29042, 29786, 33658, 34342, 38034, 38790, 42514, 43354, 47434
Offset: 0

Views

Author

Omar E. Pol, Feb 27 2015

Keywords

Comments

Also, total number of cubic ON cells after n generations in a three-dimensional structure in which A253090(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid.

Links

Crossrefs

Cf. A253090, A255048, A255601.

A072272 Number of active cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 5, 17, 5, 25, 17, 61, 5, 25, 25, 85, 17, 85, 61, 217, 5, 25, 25, 85, 25, 125, 85, 305, 17, 85, 85, 289, 61, 305, 217, 773, 5, 25, 25, 85, 25, 125, 85, 305, 25, 125, 125, 425, 85, 425, 305, 1085, 17, 85, 85, 289, 85, 425, 289, 1037, 61, 305, 305, 1037, 217, 1085, 773, 2753
Offset: 0

Views

Author

Miklos Kristof, Jul 09 2002

Keywords

Comments

Consider only the four nearest (N,S,E,W) neighbors of a cell together with the cell itself. In the next state, the state of a cell will change if an odd number of these five cells is ON. [Comment corrected by N. J. A. Sloane, Aug 25 2014]
Equivalently, a(n) is the number of ON cells at generation n of 2-D CA defined as follows: the neighborhood of a cell consists of the cell itself and the four adjacent E, W, N, S cells. A cell is ON iff an odd number of these cells was ON at the previous generation. - N. J. A. Sloane, Aug 20 2014. This is the odd-rule cellular automaton defined by OddRule 057 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
This is the Run Length Transform of A007483. - N. J. A. Sloane, Aug 25 2014
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Aug 25 2014
The partial sums are in A253908, in which the structure looks like an irregular stepped pyramid. - Omar E. Pol, Jan 29 2015
Rules 518, 550 and 582 also generate this sequence. - Robert Price, Mar 01 2016

Examples

			To illustrate a(0) = 1, a(1) = 5, a(2) = 5, a(3) = 17:
  ......................0
  .............0.......000
  .......0............0...0
  .0....000..0.0.0...00.0.00
  .......0............0...0
  .............0.......000
  ......................0
From _Omar E. Pol_, Jan 29 2015: (Start)
May be arranged into blocks of sizes A011782:
  1;
  5;
  5,17;
  5,25,17,61;
  5,25,25,85,17,85,61,217;
  5,25,25,85,25,125,85,305,17,85,85,289,61,305,217,773;
  5,25,25,85,25,125,85,305,25,125,125,425,85,425,305,1085,17,85,85,289,85,425,289,1037,
    61,305,305,1037,217,1085,773,2753;
So the right border gives A007483.
(End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
     1;
  .....
     5;
  .....
     5;
    17;
  ...........
     5,   25;
    17;
    61;
  ......................
     5,   25,  25,   85;
    17,   85;
    61;
   217;
  ...........................................
     5,   25,  25,   85,  25, 125,  85,  305;
    17,   85,  85,  289;
    61,  305;
   217;
   773;
  ..................................................................................
     5,   25,  25,   85,  25, 125,  85,  305, 25, 125, 125, 425, 85, 425, 305, 1085;
    17,   85,  85,  289,  85, 425, 289, 1037;
    61,  305, 305, 1037;
   217, 1085;
   773;
  2753;
  ...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
It appears that the configuration of ON cells of T(s,r,k) is of the same kind as the configuration of ON cells of T(s+1,r,k).
(End)

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; pp. 170-179.

Links

Crossrefs

Cf. A048883, A170878 (first differences), A253908 (partial sums).
See A253090 for 9-celled neighborhood version.

Programs

  • Maple
    C:=f->subs({x=1,y=1},f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x,y).
OddCA:=proc(f,M) global C; local n,a,i,f2,g,p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1; g:=f2;
for n from 0 to M do a:=[op(a),C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i],i=1..nops(a))]);
end;
f:=1+1/x+x+1/y+y;
OddCA(f,100);
# N. J. A. Sloane, Aug 20 2014
  • Mathematica
    Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 614, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]] (* N. J. A. Sloane, Apr 17 2010 *)
ArrayPlot /@ CellularAutomaton[{ 614, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},6] (* N. J. A. Sloane, Aug 25 2014 *)

Formula

a(0)=1; thereafter a(2t)=a(t), a(4t+1)=5*a(t), a(4t+3)=3*a(2t+1)+2*a(t). - N. J. A. Sloane, Jan 26 2015

Extensions

Extended and edited by John W. Layman, Jul 17 2002
Minor edits by N. J. A. Sloane, Jan 07 2010
More terms from N. J. A. Sloane, Apr 17 2010
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