A139251 -id:A139251 - cp's OEIS frontend

A160426 Toothpick sequence starting from an asymmetric cross, with four edges of length 1, 2, 3 and 4, formed by five toothpicks of length 2.

0, 5, 9, 17, 30, 42, 52, 69, 90, 102, 112, 129, 150, 170, 196, 237, 274, 286, 296, 313, 334, 354, 380, 421, 458, 478, 504, 545, 590, 642, 724, 829, 898, 910, 920, 937, 958, 978, 1004, 1045, 1082, 1102, 1128, 1169, 1214

Offset: 0

Omar E. Pol, May 25 2009, May 29 2009

nonn

On the infinite square grid we start at stage 0 with no toothpicks. At stage 1 we place three consecutive toothpicks and two orthogonal toothpicks, as an asymetric cross with four edges of length 1, 2, 3, and 4, then a(1)=5. At stage 2 we place 4 toothpicks. And so on...

The sequence gives the number of toothpicks in the structure after n stages. A160427 (the first differences) gives the number added at the n-th stage. See A139250 for more information about toothpick sequences.

Nathaniel Johnston, Table of n, a(n) for n = 0..455
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
Nathaniel Johnston, C program for computing terms

Cf. A139250, A139251, A160740, A160800, A160802, A160808.

Terms after a(13) from Nathaniel Johnston, Mar 31 2011

A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 8, 16, 24, 32, 40, 56, 72, 80, 88, 104, 120, 136, 160, 200, 232, 240, 248, 264, 280, 296, 320, 360, 392, 408, 432, 472, 512, 560, 640, 744, 808, 816, 824, 840, 856, 872, 896, 936, 968, 984, 1008, 1048, 1088, 1136, 1216, 1320, 1384, 1400, 1424, 1464, 1504, 1552

Offset: 0

Omar E. Pol, May 25 2009

nonn

On the infinite square grid we start at stage 0 with no toothpicks. Toothpicks have length 2. At stage 1 we place two consecutive toothpicks in the vertical direction and two consecutive toothpicks in the horizontal direction forming a cross centered at the origin. At stage 2 we place four toothpicks. At stage 3 we place eight toothpicks. For more information about the toothpick sequences see A139250. - Omar E. Pol, Nov 24 2011

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
Mathematical Association of America, NumberADay: 1136
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, A160736, A160738, A160741.

Cf. A160170, A160426, A194432, A194434.

a(n) = 4*A160406(n).

More terms from N. J. A. Sloane, May 25 2009

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.

A162796 Number of toothpicks in the toothpick structure A139250 that are orthogonal to the initial toothpick after n even rounds.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 42, 70, 86, 94, 106, 134, 154, 182, 222, 310, 342, 350, 362, 390, 410, 438, 478, 566, 602, 630, 670, 758, 814, 906, 1046, 1302, 1366, 1374, 1386, 1414, 1434, 1462, 1502, 1590, 1626, 1654, 1694, 1782, 1838, 1930, 2070, 2326, 2394, 2422, 2462

Offset: 0

Omar E. Pol, Jul 14 2009

nonn

Also, partial sums of A162794.

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, A159791, A159792, A162793, A162794, A162795, A162797.

a139251 := BFILETOLIST("b139251.txt") ; A162794 := proc(n) global a139251; op(2*n,a139251) ; end: A162796 := proc(n) add( A162794(k),k=1..n) ; end: seq(A162796(n),n=1..120) ; # R. J. Mathar, Sep 27 2009

terms = 100;
Cases[Import["https://oeis.org/A139251/b139251.txt", "Table"], {, }][[;; 2terms;; 2, 2]] // Accumulate (* Jean-François Alcover, Mar 24 2020 *)

Extended by R. J. Mathar, Sep 27 2009

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 9, 9, 7, 13, 15, 1, 3, 5, 9, 9, 9, 17, 25, 17, 7, 13, 23, 27, 19, 31, 31, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 7, 13, 23, 27, 27, 43, 67, 59, 27, 31, 55, 69, 49, 69, 63, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41

Offset: 0

N. J. A. Sloane, Jan 10 2010

Row k has 2^k terms.

Right border gives the positive terms of A000225. - Omar E. Pol, Sep 28 2013

			Triangle begins:
1;
1,3;
1,3,5,7;
1,3,5,9,9,7,13,15;
1,3,5,9,9,9,17,25,17,7,13,23,27,19,31,31;
1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,67,59,27,31,55,69,49,69,63;
...

N. J. A. Sloane, Table of n, a(n) for n = 0..1023

Cf. A169779 (partial sums).

Cf. A139250, A139251, A151723, A151724, A170899.

Equals A170905(n) - 1.

A290221 Number of elements added at n-th stage to the structure of the narrow cross described in A290220.

Original entry on oeis.org

0, 2, 4, 4, 8, 8, 8, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12, 8, 16, 12

Offset: 0

Omar E. Pol, Jul 24 2017

For n = 0..6 the sequence is similar to some toothpick sequences.
The surprising fact is that for n >= 7 the sequence has periodic tail. More precisely, it has period 3: repeat [8, 16, 12]. This tail is in accordance with the expansion of the four arms of the cross.
This is essentially the first differences of A290221. The behavior is similar to A289841 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

			For n = 0..6 the sequence is: 0, 2, 4, 4, 8, 8, 8;
Terms 7 and beyond can be arranged in a rectangular array with three columns as shown below:
8, 16, 12;
8, 16, 12;
8, 16, 12;
8, 16, 12;
...

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A004767, A008584, A042963, A139250, A139251, A220500, A220501, A289841 (a more complex cross), A290220, A294021.

LinearRecurrence[{0,0,1},{0,2,4,4,8,8,8,8,16,12},90] (* Harvey P. Dale, Dec 31 2018 *)

concat(0, Vec(2*x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^7 + 2*x^8) / ((1 - x)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, Nov 12 2017

G.f.: 2*x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^7 + 2*x^8) / ((1 - x)*(1 + x + x^2)). - Colin Barker, Nov 12 2017

A294021 Number of elements added at n-th stage to the structure of the cellular automaton described in A294020.

Original entry on oeis.org

0, 1, 4, 4, 6, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22, 8, 4, 14, 24, 16, 22

Offset: 0

Omar E. Pol, Oct 21 2017

Essentially the first differences of A294020.
The sequence starts with 0, 1, 4, 4, 6. For n >= 5 the sequence has a periodic tail. More precisely, it has period 6: repeat [8, 4, 14, 24, 16, 22]. This tail is in accordance with the expansion of the two arms of the structure.
The behavior is similar to A289841 and A290221 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence.

			The sequence begins:
0, 1, 4, 4, 6;
The periodic tail begins:
8, 4, 14, 24, 16, 22;
8, 4, 14, 24, 16, 22;
8, 4, 14, 24, 16, 22,
8, 4, 14, 24, 16, 22;
8, 4, 14, 24, 16, 22;
...

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A139251, A194271, A289841, A290221, A294020.

concat(0, Vec(x*(1 + 4*x + 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 + 13*x^6 + 20*x^7 + 12*x^8 + 16*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, Nov 11 2017

From Colin Barker, Nov 11 2017: (Start)
G.f.: x*(1 + 4*x + 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 + 13*x^6 + 20*x^7 + 12*x^8 + 16*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-6) for n > 10.
(End)

A159789 a(n) = A159786(n+1)/4.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 8, 12, 12, 13, 16, 18, 19, 26, 44, 56, 56, 57, 60, 62, 63, 70, 84, 92, 93, 98, 106, 111, 120, 152, 216, 240, 240, 241, 244, 246, 247, 254, 268, 276, 277, 282, 290, 295, 304, 336, 384, 408, 409, 414

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 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

Toothpick sequence: A139250.

Cf. A139251, A153000, A159786, A159787, A159788, A159799.

More terms from Colin Barker, Apr 19 2015

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

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

cp's OEIS Frontend

A160426 Toothpick sequence starting from an asymmetric cross, with four edges of length 1, 2, 3 and 4, formed by five toothpicks of length 2.

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A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

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A161831 First differences of A161830.

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A162796 Number of toothpicks in the toothpick structure A139250 that are orthogonal to the initial toothpick after n even rounds.

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A170898 Triangle read by rows, obtained by dividing A151724 by 6.

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A290221 Number of elements added at n-th stage to the structure of the narrow cross described in A290220.

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Mathematica

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A294021 Number of elements added at n-th stage to the structure of the cellular automaton described in A294020.

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A159789 a(n) = A159786(n+1)/4.

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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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