A160121 -id:A160121 - cp's OEIS frontend

A194441 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194440.

0, 1, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 12, 16, 8, 16, 16, 4, 4, 8, 12, 16, 16, 24, 26, 24, 12, 20, 28, 40, 20, 32, 32, 4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 12, 20, 32, 48, 48, 64, 70, 76, 30, 44, 64, 88, 44, 64, 64, 4, 4, 8, 12

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194440.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,8,8,
4,4,8,12,16,8,16,16,
4,4,8,12,16,16,24,26,24,12,20,28,40,20,32,32,
4,4,8,12,16,16,24,26,24,20,32,40,64,40,48,54,40,12,20,...
.
It appears that rows converge to A194696.

Cf. A139251, A160121, A160407, A161831, A194271, A194440, A194443, A194445, A194692, A194693, A194696.

Conjectures for n = 2^k+j, if -1<=j<=3:
a(2^k-1) = 2^k, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 2.
End of conjectures.

More terms from Omar E. Pol, Dec 28 2012

A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 33, 48, 20, 8, 16, 28, 40, 44, 50, 60, 28, 16, 32, 60, 32, 32, 49, 64, 4, 4, 8

Offset: 0

Omar E. Pol, Aug 29 2011

nonn

Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

			If written as a triangle:
0,
1,
2,
4,4,
4,4,7,8,
4,4,8,12,8,8,13,16,
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32,
4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...
.
It appears that rows converge to A194697.

Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

Conjectures for n = 2^k+j, if -6<=j<=6:
a(2^k-6) = 2^(k-2), if k >= 3.
a(2^k-5) = 2^(k-1), if k >= 3.
a(2^k-4) = 2^k-4, if k >= 2.
a(2^k-3) = 2^(k-1), if k >= 3.
a(2^k-2) = 2^(k-1), if k >= 2.
a(2^k-1) = 3*2^(k-2)+1, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 3.
a(2^k+4) = 12, if k >= 3.
a(2^k+5) = 16, if k >= 4.
a(2^k+6) = 16, if k >= 4.
End of conjectures.

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

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

A161331 First differences of A161330.

Original entry on oeis.org

0, 2, 6, 6, 6, 18, 6, 18, 18, 18, 30, 18, 30, 42, 6, 18, 18, 30, 54, 30, 78, 42, 54, 78, 42, 66, 78, 18, 42, 42, 54, 90, 66, 126, 90, 90, 102, 66, 78, 90, 90, 90, 54, 66, 114, 78, 126, 126, 102, 102, 138, 102, 162, 102, 114, 162, 126, 162, 114, 102, 102, 126, 186, 186, 150, 138, 126, 162, 162, 186, 198, 114, 114, 162

Offset: 0

Omar E. Pol, Jun 07 2009

nonn

Number of E-toothpicks added to the snowflake structure at n-th round.

David Applegate, Table of n, a(n) for n = 0..1000
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, A160173, A161328, A161329, A161330.

More terms from David Applegate, Dec 13 2012

A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

Original entry on oeis.org

0, 3, 6, 12, 12, 12, 24, 36, 24, 12, 24, 48, 60, 48, 60, 84, 48, 12, 24, 48, 60, 60, 84, 132, 132, 72, 60, 120, 168, 144, 156, 192, 96, 12, 24, 48, 60, 60, 84, 132, 132, 84, 84, 156, 228, 228, 228

Offset: 0

Omar E. Pol, Dec 07 2010

First differences of A182632.

a(n) is also the number of components added at n-th stage in the toothpick structure formed by V-toothpicks with an initial Y-toothpick, since a V-toothpick has two components and a Y-toothpick has three components (For more information see A161206, A160120, A161644).

			From _Omar E. Pol_, Feb 08 2013 (Start):
When written as a triangle:
0;
3;
6;
12,12;
12,24,36,24;
12,24,48,60,48,60, 84, 48;
12,24,48,60,60,84,132,132,72,60,120,168,144,156,192,96;
12,24,48,60,60,84,132,132,84,84,156,228,228,228,...
...
It appears that positive terms of the right border are A007283.
(End)

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
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

A139250, A139251, A160120, A160121, A161206, A161207, A161644, A161645, A182632.

It appears that a(n) = 2*A161645(n) but with a(1)=3.

a(n) = 3*A182635(n). - Omar E. Pol, Feb 09 2013

A161831 First differences of A161830.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 8, 4, 4, 4, 8, 6, 8, 10, 18, 10, 4

Offset: 1

Omar E. Pol, Jun 20 2009

Number of Y-toothpicks added to the sieve at the n-th round.

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, A160407, A161830, A161832, A161833.

A161329 First differences of A161328.

Original entry on oeis.org

1, 3, 5, 7, 13, 11, 17, 15, 21, 23, 25, 27, 33, 27, 25, 15, 25, 35, 41, 55, 53, 59, 61, 59, 65, 63, 57, 47, 37, 47, 65, 71, 97, 95, 105, 95, 89, 83, 81, 87, 93, 79, 73, 79, 89, 107, 113, 119, 113, 115, 117, 135, 125, 127, 129, 135, 153, 135

Offset: 1

Omar E. Pol, Jun 07 2009

nonn

Number of E-Toothpicks added to the E-Toothpick structure at the n-th round.

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. A139251, A160121, A160173, A161328, A161331.

a(8) corrected and more terms added by R. J. Mathar, Jan 21 2010

A161830 Y-toothpick triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 19, 27, 31, 35, 39, 47, 53, 61, 71, 89, 99, 103

Offset: 0

Omar E. Pol, Jun 20 2009

Y-toothpick sequence starting at the corner of an infinite hexagon in which its vertex touch an endpoint of the initial Y-toothpick and the two other endpoints are equidistant from the nearest sides of the hexagon.
The sequence gives the number of Y-toothpicks in the structure after n rounds. A161831 (the first differences) gives the number added at the n-th round.
See the Y-toothpick sequence A160120 for more information about the recursive, fractal-like structure.

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, A160120, A160121, A160406, A160407, A161831, A161832, A161833.

cp's OEIS Frontend

A194441 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194440.

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A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

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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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A172311 First differences of A172310.

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A160161 First differences of the 3D toothpick numbers A160160.

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PARI

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A161331 First differences of A161330.

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A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

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A161831 First differences of A161830.

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A161329 First differences of A161328.

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