A160171 -id:A160171 - cp's OEIS frontend

A160121 First differences of A160120.

1, 3, 3, 9, 3, 9, 9, 21, 9, 9, 9, 21, 15, 21, 27, 51, 27, 9, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 81, 21, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135, 141, 171, 279, 231, 69, 9, 21, 15, 21, 27

Offset: 1

Omar E. Pol, May 02 2009

nonn

Number of Y-toothpicks added at n-th stage to the Y-toothpick structure of A160120.

For a simpler version, see A151710. - Omar E. Pol, Dec 18 2012

			Contribution from _Omar E. Pol_, Jun 18 2009: (Start)
May be written as a triangle:
1,
3,
3,
9,
3,9,
9,21,9,9,
9,21,15,21,27,51,27,9,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,81,21,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279,231,69;
Rows converge to A161326.
(End)
Contribution from _Omar E. Pol_, Dec 18 2012: (Start):
Also this sequence may be written as another triangle (according to the structure of triangle A151710):
1;
3;
3,  9;
3,  9,9,21;
9,  9,9,21,15,21,27,51;
27, 9,9,21,15,21,27,51,33,21,27,51,51,57,69,117;
81,21,9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279;
(End)

JungHwan Min, Table of n, a(n) for n = 1..5000
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.]
David Applegate, The movie version
Omar E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [_Omar E. Pol_, Nov 02 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A151710, A160171, A160173, A160120, A160715, A161207, A161329, A161331, A161326, A161431, A147582.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 1, 3, in == 0 && Count[out, 1] >= 2, 2, in == 0 && Count[out, 1] == 1, 1, True, in]]; A160121[n_] := Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)
A160121[n_] := Count[CellularAutomaton[{13390417258775213635414055181254541831894674613399006361662885886563211940509571858857491972104491013971547937418035084866785430974106432144737472376143620, 4, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009

A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

Original entry on oeis.org

0, 1, 3, 5, 9, 9, 9, 13, 25, 21, 9, 13, 25, 25, 25, 37, 73, 57, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 165, 9, 13, 25, 25, 25, 37, 73, 61, 25, 37, 73, 73, 73, 109, 217, 169, 25, 37, 73, 73, 73, 109, 217, 181, 73, 109, 217, 217, 217, 325, 649, 489, 9, 13, 25

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

Essentially the first differences of A160172.
For further information see the Applegate-Pol-Sloane paper, chapter 11: T-shaped toothpicks. See also the figure 16 in the mentioned paper. - Omar E. Pol, Nov 18 2011
The numbers n in increasing order such that the triple [n, n, n] can be found here, give A199111. [Observed by Omar E. Pol, Nov 18 2011. Confirmed by Alois P. Heinz, Nov 21 2011]

			From _Omar E. Pol_, Feb 09 2010: (Start)
If written as a triangle:
0;
1;
3;
5;
9,9;
9,13,25,21;
9,13,25,25,25,37,73,57;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,165;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489;
9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109...
(End)

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

David Applegate, The movie version
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.], which is also available at arXiv:1004.3036v2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A139251, A160172, A160121, A160171, A161207, A161329, A172311.

wt[n_] := DigitCount[n, 2, 1];
a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := 2/3 (3^wt[n-1] + 3^wt[n-2]) + 1;
Table[a[n], {n, 0, 68}] (* Jean-François Alcover, Aug 18 2018, after N. J. A. Sloane *)

a(n) = (2/3)*(3^wt(n-1) + 3^wt(n-2))+1 (where wt is A000120), for n >= 3. - N. J. A. Sloane, Jan 01 2010

More terms from N. J. A. Sloane, Jan 01 2010

A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 13, 21, 45, 77, 109, 165, 245, 325, 413, 525, 685, 853, 1093, 1317, 1661, 1981, 2301, 2645, 3093, 3621, 4157, 4861, 5565, 6437, 7173, 8053, 8893, 9917, 11005, 12261, 13589, 14981, 16397, 17837, 19341, 20997

Offset: 0

Omar E. Pol, May 03 2009

nonn

Here a "X-toothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a X-toothpick can be represented by set of four connected toothpicks forming a cross.
We start at stage 0 on the Z^3 lattice with no X-toothpicks.
At stage 1 place a X-toothpick.
Rule: each exposed endpoint of the X-toothpicks of the old generation must be touched by the midpoint of a X-toothpick of new generation (see illustrations).
The sequence gives the number of X-toothpicks in the three-dimensional structure after n-th stage. A170171 (the first differences) gives the number of X-toothpicks added at n-th stage.
For a similar sequence but starting with a single toothpick see A170876.
For the X-toothpick sequence on Z^2 lattice see A147562, the Ulam-Warburton cellular automaton.
For more information about the growth of toothpicks see A139250.

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.],
R. J. Mathar, C++ program
R. J. Mathar, Views after stages 1 to 10 (PDF)
Olbaid Fractalium, Toothpick sequence Part 1, (Watch from minute 3:14), Youtube video (2023).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A160120, A160160, A160171, A170876.

Cf. A147562. - Omar E. Pol, Mar 28 2011

C++ program, illustrations and more terms (a(6)-a(38)) based on Email from R. J. Mathar dated Jan 10 2010.

A172311 First differences of A172310.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 14, 18, 18, 20, 24, 24, 38, 34, 42, 34, 26, 28, 32, 38, 52, 54, 64, 58, 68, 60, 60, 50, 66, 70, 70, 74, 50, 52, 60, 54, 64, 66, 84, 88, 116, 106, 132, 100, 136, 126, 140, 106, 118, 100, 122, 106, 138, 114, 138, 132, 152, 156, 176, 158, 190, 166, 158, 154, 98, 88, 132, 82, 124, 94, 112

Offset: 0

Omar E. Pol, Jan 31 2010

nonn

Number of L-toothpicks added to the L-toothpick structure of A172310 at the n-th stage.

David Applegate, The movie version
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, A160121, A160171, A160173, A161207, A161329, A172310.

More terms from Nathaniel Johnston, Nov 15 2010

Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

A160161 First differences of the 3D toothpick numbers A160160.

Original entry on oeis.org

0, 1, 2, 4, 8, 8, 8, 8, 16, 32, 56, 32, 16, 8, 16, 32, 56, 56, 64, 80, 152, 232, 352, 144, 48, 32, 24, 40, 56, 56, 64, 80, 152, 232, 352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488, 2000, 768, 304, 336, 264, 192, 112, 120, 128, 112, 168, 240, 352, 216, 168, 176, 272, 360, 496

Offset: 0

Omar E. Pol, May 03 2009

Number of toothpicks added at n-th stage to the three-dimensional toothpick structure of A160160.

The sequence should start with a(1) = 1 = A160160(1) - A160160(0), the initial a(0) = 0 seems purely conventional and not given in terms of A160160. The sequence can be written as a table with rows r >= 0 of length 1, 1, 1, 3, 9, 18, 36, ... = 9*2^(r-4) for row r >= 4. In that case, rows 0..3 are filled with 2^r, and all rows r >= 3 have the form (x_r, y_r, x_r) where x_r and y_r have 3*2^(r-4) elements, all multiples of 8. Moreover, y_r[1] = a(A033484(r-2)) = x_{r+1}[1] = a(A176449(r-3)) is the largest element of row r and thus a record value of the sequence. - M. F. Hasler, Dec 11 2018

			Array begins:
===================
    x     y     z
===================
          0     1
    2     4     8
    8     8     8
   16    32    56
   32    16     8
   16    32    56
   56    64    80
  152   232   352
  144    48    32
...
From _Omar E. Pol_, Feb 28 2018: (Start)
Also, starting with 1, the sequence can be written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
   1,  2,  4;
   8,  8,  8;
   8, 16, 32, 56, 32, 16;
   8, 16, 32, 56, 56, 64, 80, 152, 232, 352, 144, 48;
  32, 24, 40, 56, 56, 64, 80, 152, 232, 352, 216, 168, 176, 272, 360, 496, 448, ...
(End)
If one starts rows with a(A176449(k) = 9*2^k-2), they are of the form A_k, B_k, A_k where A_k and B_k have 3*2^k elements and the first element of A_k is the first element of B_{k-1} and the largest of that (previous) row:
   k | a(9*2^k-2, ...) = A_k ; B_k ; A_k
  ---+-------------------------------------
     | a( 1 .. 6) = (1, 2, 4, 8, 8, 8)   (One might consider a row (8 ; 8 ; 8).)
   0 | a( 7, ...) = (8, 16, 32 ; 56, 32, 16 ; 8, 16, 32)
   1 | a(16, ...) = (56, 56, 64, 80, 152, 232 ; 352, 144, 48, 32, 24, 40 ;
     |               56, 56, 64, 80, 152, 232)
   2 | a(34, ...) = (352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488 ;
     |               2000, 768, 304, 336, 264, 192, 112, 120, 128, 112, 168, 240 ;
     |               352, 216, 168, 176, 272, 360, 496, 448, 536, 664, 1168, 1488)
   3 | a(70, ...) = (2000, 984, ... ; 10576, 4304, ... ; 2000, 984, ...)
   4 | a(142, ...) = (10576, 5016, ... ; 54328, 24120, ...; 10576, 5016, ...)
  etc. - _M. F. Hasler_, Dec 11 2018

M. F. Hasler, Table of n, a(n) for n = 0..500
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 toothpick sequences

Cf. A160160, A139250, A139251, A160120, A160121, A160171, A170875.

A160161_vec(n)=(n=A160160_vec(n))-concat(0,n[^-1]) \\ M. F. Hasler, Dec 11 2018

A160161_vec(n)={local(E=[Vecsmall([1,1,1])], s(U)=[Vecsmall(Vec(V)+U)|V<-E], J=[], M, A, B, U); [if(i>4,8*#E=setminus(setunion(A=s(U=matid(3)[i%3+1,]), B=select(vecmin,s(-U))), J=setunion(setunion(setintersect(A, B), E), J)),2^(i-1))|i<-[1..n]]} \\ Returns the vector a(1..n). (A160160 is actually given as partial sums of this sequence, rather than the converse.) - M. F. Hasler, Dec 12 2018

a(9*2^k - m) = a(6*2^k - m) for all k >= 0 and 2 <= m <= 3*2^(k-1) + 2. - M. F. Hasler, Dec 12 2018

Extended to 78 terms with C++ program by R. J. Mathar, Jan 09 2010

Edited and extended by M. F. Hasler, Dec 11 2018

cp's OEIS Frontend

A160121 First differences of A160120.

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A160173 Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.

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A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

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A172311 First differences of A172310.

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A160161 First differences of the 3D toothpick numbers A160160.

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