A160173 -id:A160173 - 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

A161207 First differences of A161206.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 12, 12, 18, 24, 30, 30, 28, 30, 20, 12, 18, 26, 34, 42, 50, 56, 54, 44, 48, 64, 82, 80, 68, 66, 36, 12, 18, 26, 34, 42, 50, 58, 58, 54, 66, 90, 114, 126, 122, 120, 102, 60, 48, 70, 94, 118, 142, 160, 162, 136, 130, 160, 204, 198, 160, 142, 68, 12

Offset: 1

Omar E. Pol, Jun 08 2009

nonn

Number of V-toothpicks added to the V-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. A139250, A139251, A160121, A160173, A161206, A161329, A161331.

More terms from R. J. Mathar, Jan 21 2010

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by 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

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

A162201 First differences of A162200.

Original entry on oeis.org

0, 2, 0, 3, -1, 3, -1, 3, -3, 4, -2, 5, -1, 3, -3, 6, -2, 4, -3, 3, -2, 5, -3, 7, -4, 3, -1, 3, -1, 9, -7, 5, -2, 6, -4, 4, -4, 5, -3, 6, -2, 6, -4, 3, -1, 7, -10, 8, -1, 3, -3, 4, -4, 8, -4, 6, -2, 4, -3, 3, -4, 12, -7, 3, -1, 9, -8, 8, -4, 3, -3, 7, -5, 6, -3, 5, -5, 6, -4, 9, -4, 6, -4, 4, -3

Offset: 1

Omar E. Pol, Jun 28 2009

The absolute value of a(n) is also the length of the n-th vertical edge in the graph of the "mountain path" function for prime numbers.

See A162200 for the length of the n-th horizontal component.

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The mountain path of the primes

Cf. A000040, A006005, A008578, A162200, A162202, A162203, A162340, A162341, A162342, A162343, A162344.

A031131 := proc(n) ithprime(n+2)-ithprime(n) ; end: A052288 := proc(n) A031131(n)/2 ; end: A160173 := proc(n) if n <= 2 then 2*(n-1); elif n mod 2 = 0 then A052288(n-1) ; else 2-A052288(n-1) ; fi; end: seq(A160173(n),n=1..150) ; # R. J. Mathar, Jul 15 2009

From R. J. Mathar, Jul 15 2009: (Start)
a(n) = A052288(n-1) if n >= 2, n even.
a(n) = 2 - A052288(n-1) if n >= 3, n odd. (End)

Edited by Omar E. Pol, Jul 02 2009

More terms from R. J. Mathar, Jul 15 2009

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

A294963 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A294962.

Original entry on oeis.org

1, 4, 8, 8, 12, 20, 16, 8, 24, 16

Offset: 1

Omar E. Pol, Feb 10 2018

			The finite sequence can be written as an array of four columns as shown below:
   1,  4,  8, 8;
  12, 20, 16, 8;
  24, 16.
The first column gives the number of toothpicks of length 2.
The second column gives the number of D-toothpicks.
The third column gives the number of toothpicks of length 1.
The fourth column gives the number of T-toothpicks.
The sequence contains exactly 10 terms.

Cf. A294962.
Cf. A139251 (toothpicks), A160173 (T-toothpicks), A194701 (D-toothpicks), A220501.
For other hybrid cellular automata, see A289841, A290221, A294021, A294981.

A199111 a(n) = 8*3^n + 1.

Original entry on oeis.org

9, 25, 73, 217, 649, 1945, 5833, 17497, 52489, 157465, 472393, 1417177, 4251529, 12754585, 38263753, 114791257, 344373769, 1033121305, 3099363913, 9298091737, 27894275209, 83682825625, 251048476873, 753145430617, 2259436291849, 6778308875545, 20334926626633, 61004779879897

Offset: 0

Vincenzo Librandi, Nov 03 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A005051, A160173. - Omar E. Pol, Nov 18 2011

Cf. A198644.

Magma
```
[8*3^n+1: n in [0..30]];
```

a(n) = 3*a(n-1) - 2.
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: (9-11*x)/((1-x)*(1-3*x)). - Bruno Berselli, Nov 03 2011
a(n) = 8*A000244(n) + 1 = A005051(n) + 1. - Omar E. Pol, Nov 18 2011
From Elmo R. Oliveira, May 07 2025: (Start)
E.g.f.: exp(x)*(8*exp(2*x) + 1).
a(n) = A198644(n) + 2. (End)

A299771 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A299770.

Original entry on oeis.org

1, 4, 8, 8, 12, 16, 16, 8, 24, 8

Offset: 1

Omar E. Pol, Mar 20 2018

The word of this cellular automaton is abcd. For more information see A296612.

			The finite sequence can be written as an array of four columns as shown below:
   1,  4,  8, 8;
  12, 16, 16, 8;
  24,  8.
The first column gives the number of toothpicks of length 2.
The second column gives the number of D-toothpicks of length sqrt(2).
The third column gives the number of toothpicks of length 1.
The fourth column gives the number of T-toothpicks.
The sequence contains exactly 10 terms.

Very similar to A294963.
Cf. A299770, A296612.
Cf. A139251 (toothpicks), A160173 (T-toothpicks), A194701 (D-toothpicks), A220501.
For other hybrid cellular automata, see A289841, A290221, A294021, A294981.

cp's OEIS Frontend

A160121 First differences of A160120.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A161207 First differences of A161206.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160172 T-toothpick sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A172311 First differences of A172310.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A161331 First differences of A161330.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A162201 First differences of A162200.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A161329 First differences of A161328.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A294963 a(n) is the number of elements added at n-th stage in the structure of the finite cellular automaton of A294962.

Views

Author

Keywords

Examples

Links

Crossrefs

A199111 a(n) = 8*3^n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs