A147610 -id:A147610 - cp's OEIS frontend

A151784 a(n) = 6^(wt(n) - 1) where wt(n) = A000120(n).

1, 1, 6, 1, 6, 6, 36, 1, 6, 6, 36, 6, 36, 36, 216, 1, 6, 6, 36, 6, 36, 36, 216, 6, 36, 36, 216, 36, 216, 216, 1296, 1, 6, 6, 36, 6, 36, 36, 216, 6, 36, 36, 216, 36, 216, 216, 1296, 6, 36, 36, 216, 36, 216, 216, 1296, 36, 216, 216, 1296, 216, 1296, 1296, 7776, 1, 6, 6, 36, 6, 36, 36

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,6;
  1,6,6,36;
  1,6,6,36,6,36,36,216;
  1,6,6,36,6,36,36,216,6,36,36,216,36,216,216,1296;
  1,6,6,36,6,36,36,216,6,36,36,216,36,216,216,1296,6,36,36,216,36,216,216,...
(End)

Cf. A000120, A048881, A048896, A147610, A151780, A151783, A000120.

Cf. A000400. - Omar E. Pol, Jul 21 2009

a(n) = 6^(hammingweight(n)-1); \\ Michel Marcus, Nov 15 2022

A183061 First differences of A183060.

Original entry on oeis.org

0, 1, 3, 3, 7, 3, 7, 7, 19, 3, 7, 7, 19, 7, 19, 19, 55, 3, 7, 7, 19, 7, 19, 19, 55, 7, 19, 19, 55, 19, 55, 55, 163, 3, 7, 7, 19, 7, 19, 19, 55, 7, 19, 19, 55, 19, 55, 55, 163, 7, 19, 19, 55, 19, 55, 55, 163, 19, 55, 55, 163, 55, 163, 163, 487, 3

Offset: 0

Omar E. Pol, Feb 20 2011

nonn

The sequence gives the number of cells turned "ON" at the n-th stage in the structure of A183060.

			If written as a triangle begins:
0,
1,
3,
3,7,
3,7,7,19,
3,7,7,19,7,19,19,55,
3,7,7,19,7,19,19,55,7,19,19,55,19,55,55,163,
It appears that row sums give A007582.
It appears that last terms of rows give A100702.

David 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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A007582, A100702, A139250, A139251, A147582, A147610, A183060.

a(n) = 1 + A147582(n)/2.

a(n) = 1 + 2*A147610(n).

A183149 Number of toothpicks added at n-th stage to the toothpick structure of A183148.

Original entry on oeis.org

0, 1, 3, 9, 9, 21, 9, 21, 21, 57, 9, 21, 21, 57, 21, 57, 57, 165, 9, 21, 21, 57, 21, 57, 57, 165, 21, 57, 57, 165, 57, 165, 165, 489, 9, 21, 21, 57, 21, 57, 57, 165, 21, 57, 57, 165, 57, 165, 165, 489, 21, 57, 57, 165, 57, 165

Offset: 0

Omar E. Pol, Mar 28 2011, Apr 02 2011

nonn

Essentially the first differences of A183148.

			If written as a triangle begins:
0,
1,
3,
9,
9,21,
9,21,21,57,
9,21,21,57,21,57,57,165,
9,21,21,57,21,57,57,165,21,57,57,165,57,165,165,489,

David 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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A147582, A147610, A160173, A161411, A183061, A183127, A183148.

a(n) = 3*A183061(n-1), for n >=2

A322663 First differences of A322662 divided by 12.

Original entry on oeis.org

1, 1, 7, 1, 6, 11, 14, 3, 11, 14, 25, 5, 18, 21, 37, 4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70, 10, 42, 41, 81, 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, 18, 37, 64, 151, 41, 80, 126, 131, 29

Offset: 1

Bradley Klee, Dec 22 2018

nonn

Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.

			Written as a 2^k triangle:
1,
1, 7,
1, 6,  11, 14,
3, 11, 14, 25, 5,  18, 21, 37,
4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70,  10, 42, 41, 81,
4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...

Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.

HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
MoveSet2 =Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
Clear@Pts;Pts[0] = {{0, 0}};
Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet2],1]],{x_,1}:>x]]];
Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0,32],2,1]]

a(n) = (A322662(n)-A322662(n-1))/12.

cp's OEIS Frontend

A151784 a(n) = 6^(wt(n) - 1) where wt(n) = A000120(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A183061 First differences of A183060.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A183149 Number of toothpicks added at n-th stage to the toothpick structure of A183148.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A322663 First differences of A322662 divided by 12.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula