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.

A079315 Number of cells that change from OFF to ON at stage n of the cellular automaton described in A079317.

Original entry on oeis.org

0, 1, 4, 4, 12, 4, 20, 12, 44, 4, 52, 12, 76, 12, 100, 36, 172, 4, 180, 12, 204, 12, 228, 36, 300, 12, 324, 36, 396, 36, 468, 108, 684, 4, 692, 12, 716, 12, 740, 36, 812, 12, 836, 36, 908, 36, 980, 108, 1196, 12, 1220, 36, 1292, 36, 1364, 108, 1580, 36, 1652, 108, 1868
Offset: 0

Views

Author

N. J. A. Sloane, Feb 12 2003

Keywords

Comments

Start with cell (0,0) ON; at each succeeding stage the cells that share exactly one edge with an active cell change their state.
This is not the CA discussed by Singmaster in the reference given in A079314. That was an error based on my misreading of the paper. - N. J. A. Sloane, Aug 05 2009

References

  • D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.

Links

Crossrefs

Cf. A079316-A079319, A147582, A151914, A151917, A151920, A151921, A139250.

Programs

  • Mathematica
    wt[n_] := DigitCount[n, 2, 1];
A147582[n_] := If[n == 1, 1, 4*3^(wt[n-1]-1)];
A151914[n_] := Switch[n, 0, 0, 1, 4, _, (8/3)*Sum[3^wt[i], {i, 1, n-1}]+4];
a[n_] := If[OddQ[n], A147582[(n-1)/2+1], A151914[n/2]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 21 2024 *)

Formula

a(2n+1) = A147582(n+1), a(2n) = A151914(n).

Extensions

More terms from John W. Layman, Oct 30 2003
Edited by N. J. A. Sloane, Aug 05 2009

A151917 a(0)=0, a(1)=1; for n>=2, a(n) = (2/3)*(Sum_{i=1..n-1} 3^wt(i)) + 1, where wt() = A000120().

Original entry on oeis.org

0, 1, 3, 5, 11, 13, 19, 25, 43, 45, 51, 57, 75, 81, 99, 117, 171, 173, 179, 185, 203, 209, 227, 245, 299, 305, 323, 341, 395, 413, 467, 521, 683, 685, 691, 697, 715, 721, 739, 757, 811, 817, 835, 853, 907, 925, 979, 1033, 1195, 1201, 1219
Offset: 0

Views

Author

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Keywords

Comments

Also, total number of "ON" cells at n-th stage in two of the four wedges of the "Ulam-Warburton" two-dimensional cellular automaton of A147562, but including the central ON cell. It appears that this is very close to A139250, the toothpick sequence. - Omar E. Pol, Feb 22 2015

Examples

			n=3: (2/3)*(3^1+3^1+3^2+3^1) + 1 = (2/3)*18 + 1 = 13.

Links

Crossrefs

Cf. A000120, A007583, A139250, A147562, A151914, A151920.

Programs

  • Mathematica
    Array[(2/3) Sum[3^(Total@ IntegerDigits[i, 2]), {i, # - 1}] + 1 &, 50] (* Michael De Vlieger, Nov 01 2022 *)
  • PARI
    a(n) = if (n<2, n, 1 + 2*sum(i=1,n-1, 3^hammingweight(i))/3); \\ Michel Marcus, Feb 22 2015

Formula

a(n) = A151914(n)/4.
a(n) = A079315(2n)/4.
For n>=2, a(n) = 2*A151920(n-2) + 1.
For n>=1, a(n) = (1 + A147562(n))/2. - Omar E. Pol, Mar 13 2011
a(2^k) = A007583(k), if k >= 0. - Omar E. Pol, Feb 22 2015

A151921 Net gain in number of ON cells at stage n of the cellular automaton described in A079317.

Original entry on oeis.org

0, 1, 4, 0, 12, -8, 20, -8, 44, -40, 52, -40, 76, -64, 100, -64, 172, -168, 180, -168, 204, -192, 228, -192, 300, -288, 324, -288, 396, -360, 468, -360, 684, -680, 692, -680, 716, -704, 740, -704, 812, -800, 836, -800, 908, -872, 980, -872, 1196
Offset: 0

Views

Author

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Keywords

Comments

Start with cell (0,0) ON; at each succeeding stage the cells that share exactly one edge with an active cell change their state.

Links

Crossrefs

Cf. A079317, A079315, A139250, A151914, A147582.

Formula

If n is even, a(n) = A079315(n) = A151914(n/2); if n is odd, a(n) = A147582((n+1)/2) - A151914((n-1)/2).
First differences of A079317.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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