A160406 -id:A160406 - cp's OEIS frontend

A160407 First differences of toothpick numbers A160406.

1, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 4, 6, 10, 8, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 2, 2, 4, 4, 4, 6, 10, 8, 4, 6, 10, 10, 12, 20, 26, 16, 4, 6, 10, 10, 12, 20, 26, 18, 12, 20, 28, 30, 42

Offset: 1

Omar E. Pol, May 23 2009

nonn

Number of toothpicks added at n-th stage in the toothpick structure of A160406.
From Omar E. Pol, Mar 15 2020: (Start)
The cellular automaton described in A160406 has word "ab", so the structure of this triangle is as follows:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...
This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks that are parallel to initial toothpick, and the even-indexed columns (b) contain numbers of the toothpicks that are orthogonal to the initial toothpick.
For further information about the "word" of a cellular automaton see A296612. (End)

			From _Omar E. Pol_, Jul 18 2009, Mar 15 2020: (Start)
If written as a triangle:
1,1;
2,2;
2,2,4,4;
2,2,4,4,4,6,10,8;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16;
2,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,20,28,30,42;...
(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. [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. A011782, A139250, A139251, A153000, A153006, A152980, A160406, A161830, A161831, A296612.

More terms from N. J. A. Sloane, Jul 17 2009

A160736 Toothpick sequence starting from a right angle formed by 2 toothpicks: a(n)=A160406(n)*2.

Original entry on oeis.org

0, 2, 4, 8, 12, 16, 20, 28, 36, 40, 44, 52, 60, 68, 80, 100, 116, 120, 124, 132, 140, 148, 160, 180, 196, 204, 216, 236, 256, 280, 320, 372, 404, 408, 412, 420, 428, 436, 448, 468, 484, 492, 504, 524, 544, 568

Offset: 0

Omar E. Pol, May 25 2009

nonn

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, A160737, A160738, A160740.

Terms after a(9) from Nathaniel Johnston, Mar 31 2011

A160738 Toothpick sequence starting from a T formed by 3 toothpicks: a(n)=A160406(n)*3.

Original entry on oeis.org

0, 3, 6, 12, 18, 24, 30, 42, 54, 60, 66, 78, 90, 102, 120, 150, 174, 180, 186, 198, 210, 222, 240, 270, 294, 306, 324, 354, 384, 420, 480, 558, 606, 612, 618, 630, 642, 654, 672, 702, 726, 738, 756, 786, 816, 852, 912, 990, 1038, 1050, 1068, 1098, 1128, 1164

Offset: 0

Omar E. Pol, May 25 2009, Jun 19 2009

nonn

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, A160736, A160737, A160739, A160740.

More terms from R. J. Mathar, Jul 28 2009

A170886 Similar to A160406, but always staying outside the wedge, starting at stage 1 with a toothpick whose midpoint touches the vertex of the wedge.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 17, 23, 29, 37, 49, 55, 63, 75, 93, 107, 121, 141, 161, 167, 175, 187, 205, 219, 235, 259, 285, 301, 325, 363, 409, 447, 489, 541, 577, 583, 591, 603, 621, 635, 651, 675, 701, 717, 741, 779, 825, 863, 907, 963, 1005, 1021, 1045, 1083, 1129

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

See A170887 for the first differences.

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, A170887, A170888, A170890, A170892.

Terms beyond a(9) from R. J. Mathar, Jan 25 2010

A170888 Similar to A160406, but always staying outside the wedge, starting at stage 0 with a vertical half-toothpick which protrudes from the vertex of the wedge.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 59, 69, 81, 101, 127, 143, 147, 153, 163, 173, 185, 205, 231, 249, 261, 281, 309, 339, 381, 445, 511, 543, 547, 553, 563, 573, 585, 605, 631, 649, 661, 681, 709, 739, 781, 845, 911, 945, 957, 977, 1005, 1035, 1077, 1141

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

See A170889 for the first differences.

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, A170886, A170889, A170890, A170892.

Terms beyond a(10) from R. J. Mathar, Jan 25 2010

A170890 Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 14, 21, 29, 37, 43, 53, 61, 71, 83, 103, 123, 139, 151, 165, 173, 183, 195, 215, 235, 253, 271, 295, 317, 345, 385, 441, 493, 531, 559, 581, 589, 599, 611, 631, 651, 669, 687, 711, 733, 761, 801, 857, 909, 949, 983, 1015, 1037, 1065, 1105, 1161

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

The initial half-tookpick makes an angle of 90 degrees w.r.t. the wedge's direction. This breaks the symmetry and explains the changing parity of the terms. - M. F. Hasler, Jan 29 2013

			From _M. F. Hasler_, Jan 29 2013: (Start)
The first steps are illustrated as follows, where two vertical "|" or three horizontal "_" correspond to one single full toothpick:
:                                ___ ___  |___ ___|
:                 ___    |___|    |___|   | |___| |
:   _      |_      |_    | |_|    | |_|   | | |_|
:   /\     |/\     |/\     |/\   ¯¯¯|/\   |¯¯¯|/\
:  /  \    /  \    /  \    /  \     /  \      /  \
:
: a(0)=0, a(1)=1, a(2)=2, a(3)=4, a(5)=7, a(6)=10, ... (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. [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

See A170891 for the first differences.

Cf. A139250, A170886, A170888, A170892.

A170890(n, print_all=0)={ my( cnt=n>0, ee=[[1,1]], p=Set(vector(2*n-cnt,k,k-n-abs(k-n)*I)), c, d); for(i=2, n, print_all & print1(cnt","); p=setunion(p, Set(Mat(ee~)[, 1])); my(ne=[]); for(k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee); cnt} \\ - M. F. Hasler, Jan 29 2013

a(9) corrected by Omar E. Pol, following an observation by Kevin Ryde, Jan 29 2013

Terms beyond a(9) from M. F. Hasler, Jan 29 2013

A170892 Toothpick sequence similar to A160406, but always staying outside the wedge, starting at stage 1 with a vertical toothpick whose endpoint touches the vertex of the wedge.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 16, 24, 34, 44, 48, 56, 66, 78, 90, 112, 138, 156, 160, 168, 178, 190, 202, 224, 250, 270, 282, 304, 332, 364, 406, 472, 538, 572, 576, 584, 594, 606, 618, 640, 666, 686, 698, 720, 748, 780, 822, 888, 954, 990, 1002, 1024, 1052, 1084, 1126, 1192, 1260, 1308, 1350, 1418, 1502, 1604, 1750, 1944

Offset: 0

Omar E. Pol, Jan 09 2010

nonn

See A170893 for the first differences.

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, A160406, A170886, A170888, A170890, A170893.

A170892(n, print_all=0)={my( ee=[[2*I, I]], p=Set( concat( vector( 2*n-(n>0),k,k-n-abs(k-n)*I ), I )), cnt=2); print_all & print1("1,2"); n<3 & return(n); for(i=3, n, p=setunion(p, Set(Mat(ee~)[, 1])); my(c, d, ne=[]); for( k=1, #ee, setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne, Set([[c, d]])); setsearch(p, c-2*d) || ne=setunion(ne, Set([[c-2*d, -d]]))); forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1))); cnt+=#ee; print_all & print1(","cnt)); cnt} \\ - M. F. Hasler, Jan 30 2013

Terms beyond a(10) from M. F. Hasler, Jan 30 2013

A160718 a(n) = A160406(n+2)/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 17, 20, 25, 29, 30, 31, 33, 35, 37, 40, 45, 49, 51, 54, 59, 64, 70, 80, 93, 101, 102, 103, 105, 107, 109, 112, 117, 121, 123, 126, 131, 136, 142, 152, 165, 173, 175, 178, 183, 188, 194, 204, 217, 226, 232, 242, 256, 271, 292

Offset: 0

Omar E. Pol, Jun 10 2009

nonn

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, A152998, A153000, A160406, A160407, A160719.

a(n) = A160719(n) + 1 = -a(n-1) + n + (A139250(n+1) + 5)/4 for n > 0. - Jinyuan Wang, Mar 04 2020

More terms from Jinyuan Wang, Mar 04 2020

A160719 a(n) = A160406(n+2)/2 - 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 19, 24, 28, 29, 30, 32, 34, 36, 39, 44, 48, 50, 53, 58, 63, 69, 79, 92, 100, 101, 102, 104, 106, 108, 111, 116, 120, 122, 125, 130, 135, 141, 151, 164, 172, 174, 177, 182, 187, 193, 203, 216, 225, 231, 241, 255, 270, 291

Offset: 0

Omar E. Pol, Jun 10 2009

nonn

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, A153000, A160406, A160407, A160718.

More terms from Jinyuan Wang, Mar 04 2020

A170894 Similar to A160406, always staying outside the wedge, but starting with a horizontal toothpick whose endpoint touches the vertex of the wedge.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 13, 19, 27, 33, 37

Offset: 0

Omar E. Pol, Jan 09 2010

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

cp's OEIS Frontend

A160407 First differences of toothpick numbers A160406.

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A160736 Toothpick sequence starting from a right angle formed by 2 toothpicks: a(n)=A160406(n)*2.

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Extensions

A160738 Toothpick sequence starting from a T formed by 3 toothpicks: a(n)=A160406(n)*3.

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A170886 Similar to A160406, but always staying outside the wedge, starting at stage 1 with a toothpick whose midpoint touches the vertex of the wedge.

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Author

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Comments

Links

Crossrefs

Extensions

A170888 Similar to A160406, but always staying outside the wedge, starting at stage 0 with a vertical half-toothpick which protrudes from the vertex of the wedge.

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Author

Keywords

Comments

Links

Crossrefs

Extensions

A170890 Toothpick sequence similar to A160406, but always staying outside the wedge, starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.

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Comments

Examples

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Extensions

A170892 Toothpick sequence similar to A160406, but always staying outside the wedge, starting at stage 1 with a vertical toothpick whose endpoint touches the vertex of the wedge.

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Author

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Comments

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Crossrefs

Programs

PARI

Extensions

A160718 a(n) = A160406(n+2)/2.

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A160719 a(n) = A160406(n+2)/2 - 1.

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A170894 Similar to A160406, always staying outside the wedge, but starting with a horizontal toothpick whose endpoint touches the vertex of the wedge.

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Author

Keywords