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 1-5 of 5 results.

A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

Original entry on oeis.org

0, 1, 4, 4, 4, 12, 4, 4, 12, 12, 12, 36, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108, 324, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

Keywords

Comments

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n.

Examples

			From _Omar E. Pol_, Apr 02 2018: (Start)
Note that this sequence also can be written as an irregular triangle read by rows in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
0,1, 4;
4,4,12;
4,4,12,12,12,36;
4,4,12,12,12,36,12,12,36,36,36,108;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,324;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,... (End)

References

  • S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

Links

Crossrefs

Cf. A151904, A151905, A151907, A139250, A151895, A151896.

Programs

  • Maple
    f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k,j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;
A151906 := proc(n);
if (n=0) then 0;
elif (n=1) then 1;
else 8*A151905(n) + 4;
fi;
end;
  • Mathematica
    wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
A151905[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
a[n_] := Switch[n, 0, 0, 1, 1, _, 8 A151905[n] + 4];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

Formula

The three trisections are essentially A147582, A147582 and 3*A147582 respectively. More precisely, For t >= 1, a(3t) = a(3t+1) = A147582(t+1) = 4*3^(wt(t)-1), a(3t+2) = 4*A147582(t+1) = 4*3^wt(t). See A151907 for explanation.

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 149, 161, 173, 185, 213, 233, 261, 297, 333, 385, 429, 481, 533, 545, 573, 601, 629, 673, 717, 761, 837, 905, 989, 1033, 1085, 1145, 1197, 1257, 1309, 1337, 1397, 1457, 1525, 1625, 1669
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jul 30 2009

Keywords

Comments

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and
b) Q's outer squares were not considered (that is, satisfied a)) in any previous generation, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
Originally constructed in an attempt to explain the Holladay-Ulam CA shown in Fig. 2 of the 1962 Ulam article. However, as explained on page 222 of that article, the actual rule for that CA (see A151906, A151907) is different from ours.
A170896 and A267190 are also closely related cellular automata.
A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - David Applegate, Jan 30 2016

References

  • D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191.

Links

Crossrefs

See A170896, A170897 for the original Schrandt-Ulam version.
Cf. A151896 (the first differences), A139250, A151905, A151906, A151907, A267190, A267191.

Formula

We do not know of a recurrence or generating function.

Extensions

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016

A151907 Partial sums of A151906.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 33, 45, 57, 69, 105, 109, 113, 125, 137, 149, 185, 197, 209, 245, 281, 317, 425, 429, 433, 445, 457, 469, 505, 517, 529, 565, 601, 637, 745, 757, 769, 805, 841, 877, 985, 1021, 1057, 1165, 1273, 1381, 1705, 1709, 1713, 1725, 1737, 1749, 1785, 1797
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

Keywords

Comments

a(n) is the total number of ON cells after n generations in the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article.
The definition of this CA given by Ulam on pp. 216, 222 is complicated (and incomplete). However, the same structure can be obtained as follows. Take the CA of A147562 but replace each square by a Maltese cross, a cluster of five adjacent squares:
..X..
.XXX.
..X..
We guess that this was how Holladay and Ulam originally constructed the CA.
This construction corresponds to the fact that the three trisections of the difference sequence A151906 are essentially A147582, A147582 and 3*A147582 respectively.

References

  • S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

Links

Crossrefs

Cf. A151904, A151905, A151906, A139250, A151895, A151896.

Formula

The definition in terms of A151906 provides an explicit formula for a(n).

A151904 a(n) = (3^(wt(k)+f(j))-1)/2 if n = 6k+j, 0 <= j < 6, where wt = A000120, f = A151899.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40
Offset: 0

Views

Author

N. J. A. Sloane, Jul 31 2009

Keywords

Links

Crossrefs

Cf, A151899, A151902, A151905-A151907.

Programs

  • Maple
    f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
  • Mathematica
    wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
a[n_] := (3^A151902[n] - 1)/2;
Table[a[n], {n, 0, 82}] (* Jean-François Alcover, Feb 16 2023 *)
  • PARI
    a(n)=(3^(hammingweight(n\6)+[0,0,1,1,1,2][n%6+1])-1)/2 \\ Charles R Greathouse IV, Sep 26 2015

Formula

a(n) = (3^A151902(n)-1)/2.

A151899 Period 6: repeat [0, 0, 1, 1, 1, 2].

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jul 31 2009

Keywords

Links

Crossrefs

Cf. A151902, A151904, A151905, A151906, A151907.

Programs

  • Magma
    [Abs( ((1-n) mod 3) - ((1+n) mod 2) ) : n in [0..100]]; // Wesley Ivan Hurt, Aug 20 2014
  • Maple
    f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
A151899:=n->[0, 0, 1, 1, 1, 2][(n mod 6)+1]: seq(A151899(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016
  • Mathematica
    Table[Abs[Mod[-n + 1, 3] - Mod[n + 1, 2]], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 20 2014 *)
CoefficientList[Series[(x^2 + x^3 + x^4 + 2 x^5)/(1 - x^6), {x, 0, 100}], x] (* Wesley Ivan Hurt, Aug 20 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 1},{0, 0, 1, 1, 1, 2},105] (* Ray Chandler, Aug 26 2015 *)
  • PARI
    a(n)=[0,0,1,1,1,2][n%6+1]; \\ Joerg Arndt, Aug 25 2014

Formula

a(n) = 5/6 - cos(Pi*n/3)/3 - sin(Pi*n/3)/sqrt(3) - cos(2*Pi*n/3)/3 - sin(2*Pi*n/3)/sqrt(3) - (-1)^n/6. - R. J. Mathar, Oct 08 2011
G.f.: (x^2+x^3+x^4+2*x^5)/(1-x^6); a(n) = abs( mod(1-n,3) - mod(1+n,2) ). - Wesley Ivan Hurt, Aug 20 2014
a(n) = a(n-6) for n>5. - Wesley Ivan Hurt, Jun 20 2016
Showing 1-5 of 5 results.

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