A187210 -id:A187210 - cp's OEIS frontend

A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 59, 71, 79, 93, 105, 117, 121, 133, 141, 153, 165, 177, 181, 193, 201, 209, 213, 217, 221, 237, 253, 285, 318, 350, 354, 358, 362, 400, 439, 479, 483, 491, 499, 527, 543, 559, 563, 575, 583, 591, 595, 599, 603

Offset: 0

Omar E. Pol, Apr 14 2012

nonn

First differs from A000217 at a(11). For n >= 13 the Q-toothpick structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. The behavior seems to be as modular arithmetic but in a growing structure.

			Consider the illustration of the nodes in structure of A211000:
-----------------------------------------------------
After 9 stages    After 10 stages    After 11 stages
-----------------------------------------------------
.
.    1                 1                  1
.  0   2             0   2              0   2
.        3                 3                  3
.          4                 4                  4
.        5                 5                  5
.      6                 6                  6
.        7                 7                 11
.          8            10   8             10   8
.        9                 9                  9
.
----------------------------------------------------
We can see that:
a(9)  = 0+1+2+3+4+5+6+7+8+9 = a(8)+9 = 45
a(10) = 0+1+2+3+4+5+6+7+8+9+10 = a(9)+10 = 55
a(11) = 0+1+2+3+4+5+6+8+9+10+11 = a(10)-7+11 = 59

Paolo Xausa, Table of n, a(n) for n = 0..9999 (terms 0..4999 from Carole Dubois)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000217, A187210, A211000-A211003, A211010, A211011, A211020-A211023.

A211024[nmax_]:=Module[{ep={0,0},node=Association[],angle=3/4Pi,turn=Pi/2},Join[{0},Table[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];ep=AngleVector[ep,{Sqrt[2],angle}];node[ep]=n+1;Total[node],{n,0,nmax-1}]]];
A211024[100] (* Paolo Xausa, Jan 16 2023 *)

A211022 Value on the axis "y" of the center of the n-th circle formed in the structure of A211000.

Original entry on oeis.org

-6, -4, -2, 0, 2, 6, 8, 4, -10, -12, -8, 10, 12, 14, 16, 20, 22, 24, 26, 30, 32, 34, 36, 28, 38, 40, 42, 46, 48, 52, 56, 58, 60, 62, 54, 50, 44, 64, 66, 68, 18, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 106, 104, 100

Offset: 1

Omar E. Pol, Mar 31 2012

sign

The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. Values on the axis "x" are equal to 3.

Cf. A187210, A211000, A211001-A211003, A211010, A211011, A211020, A211021, A211023, A211024.

A267694 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,1). The endpoints of the left hand Q-toothpick are at (0,1) and (1,2). The endpoints of the right hand Q-toothpick are at (1,0) and (2,1). With a(0) = 0.

Original entry on oeis.org

0, 2, 5, 9, 16, 20, 27, 39, 54, 58, 65, 77, 92, 104, 127, 163, 194, 198, 205, 217, 232, 244, 267, 303, 334, 346

Offset: 0

Omar E. Pol, Jan 23 2016

a(n) is the total number of Q-toothpicks in the structure after n-th stage.
A267695 (the first differences) gives the number of Q-toothpicks added at n-th stage.
a(n) is also the total number of Q-toothpicks minus 1 after n+2 stages in the SW quadrant of the Q-toothpick structure of A187210, but with origing at (1,1) instead (0,0), and assuming that the initial Q-toothpick is in the first quadrant with origin at (0,0), if n>=1 . For more information see the comments and the formula of A187210.

Cf. A139250, A187210, A187220, A267695, A267698.

A267698 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,3) and (3,1) respectively. The endpoints of the left hand Q-toothpick are at (0,3) and (1,4). The endpoints of the right hand Q-toothpick are at (3,0) and (4,1). With a(0) = 0.

Original entry on oeis.org

0, 2, 6, 13, 25, 32, 44, 59, 79, 86, 98, 113, 133, 148, 176, 215, 251, 258, 270, 285, 305, 320, 348, 387, 423, 438

Offset: 0

Omar E. Pol, Jan 23 2016

a(n) is the total number of Q-toothpicks in the structure after n-th stage.
A267699 (the first differences) gives the number of Q-toothpicks added at n-th stage.
Note that this sequence is also related with the structure and the formula of A187210. For more information see A187210, A267694 and A139250.

Cf. A139250, A187210, A187220, A267694, A267699.

A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 25, 26, 27, 28, 28

Offset: 0

Paolo Xausa, Sep 28 2022

nonn

See A211000 for additional information.

For the definition of Q-toothpicks, see A187210.

			In the following diagrams the A211000 structure is shown at the end of the n-th stage (Q-toothpicks are depicted as straight lines instead of circle arcs).
.
n       0       1      10      15      32      39      60      65
a(n)    0       1      10      15      16      20      23      28
.
                                                                /\
                                                                \/
                                                                 \
                                                         /       /
                                                /       /\      /\
                                                \       \/      \/
              /       /\      /\      /\      /\/\    /\/\    /\/\
                        \       \       \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                         /      \/      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
.

Cf. A187210, A211000, A355479.

A357434[nmax_]:=Module[{a={0},tp={},ep1={0,0},ep2,angle=0,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];tp=Union[tp,{{ep1,ep2=AngleVector[ep1,angle]}}];ep1=ep2;AppendTo[a,Length[tp]],{n,0,nmax-1}];a];
A357434[100]

A357434(nmax) = my(a=List([0,1]), newtp=[[0, 0], [1, 1]], tp=Set([newtp]), turn=1, p1, p2); if(nmax==0, return([0]));for(n=1, nmax-1, p1=newtp[1]; p2=newtp[2]; if(isprime(n), newtp=[p2, [2*p2[1]-p1[1], 2*p2[2]-p1[2]]], if(n>5 && isprime(n-1), turn*=-1); newtp=[p2, [p2[1]-turn*(p1[2]-p2[2]), p2[2]+turn*(p1[1]-p2[1])]]); tp=setunion(tp, [newtp]); listput(a,length(tp))); Vec(a);
A357434(100)

from sympy import isprime
def A357434(nmax):
    newtp, a, turn = ((0, 0), (1, 1)), [0, 1], 1
    tp = {newtp}
    for n in range(1, nmax):
        p1, p2 = newtp[0], newtp[1]
        if isprime(n): # Continue straight
            newtp = (p2, (2*p2[0]-p1[0], 2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            newtp = (p2, (p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))
        tp.add(newtp)
        a.append(len(tp))
    return a[:nmax+1]
print(A357434(100))

A267695 First differences of A267694.

Original entry on oeis.org

0, 2, 3, 4, 7, 4, 7, 12, 15, 4, 7, 12, 15, 12, 23, 36, 31, 4, 7, 12, 15, 12, 23, 36, 31, 12

Offset: 0

Omar E. Pol, Apr 02 2016

Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A267694.

			When the positive terms are written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
2;
3;
4, 7;
4, 7, 12, 15;
4, 7, 12, 15, 12, 23, 36, 31;
4, 7, 12, 15, 12, 23, 36, 31, 12,...

Cf. A011782, A187210, A187211, A267694, A267699.

A267699 First differences of A267698.

Original entry on oeis.org

0, 2, 4, 7, 12, 7, 12, 15, 20, 7, 12, 15, 20, 15, 28, 39, 36, 7, 12, 15, 20, 15, 28, 39, 36, 15

Offset: 0

Omar E. Pol, Apr 02 2016

Number of Q-toothpicks added at n-th stage in the Q-toothpick structure of A267698.

			When the positive terms are written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
2;
4;
7, 12;
7, 12, 15, 20;
7, 12, 15, 20, 15, 28, 39, 36;
7, 12, 15, 20, 15, 28, 39, 36, 15,...

Cf. A011782, A187210, A187211, A267695, A267698.

A187216 Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.

Original entry on oeis.org

0, 2, 8, 16, 30, 52, 82, 104, 142, 196, 266, 288, 326, 380, 450, 504, 606, 756, 890, 912, 950, 1004, 1074, 1128, 1230, 1380, 1514, 1568, 1670, 1820, 1986, 2168, 2494, 2900, 3162, 3184, 3222, 3276, 3346, 3400, 3502, 3652, 3786, 3840, 3942, 4092, 4258, 4440

Offset: 0

Omar E. Pol, Mar 30 2011

nonn

The sequence gives the number of Q-toothpicks in the structure after n-th stage.
A187217 (the first differences) gives the number of Q-toothpicks added at n-th stage.
Note that in the Q-toothpick structure sometimes there is also an internal growth of Q-toothpicks.
For more information see A187210.

			On the infinite square grid at stage 0 we start with no Q-toothpicks.
At stage 1 we place two opposite Q-toothpicks centered at (0,0). One of the Q-toothpicks lies on the first quadrant with its endpoints at (0,1) and (1,0). The other Q-toothpick lies on the third quadrant with its endpoints at (0,-1) and (-1,0). So a(1) = 2. There are 4 exposed endpoints.
At stage 2 we place 6 Q-toothpicks, so a(2) = 2+6 = 8.
At stage 3 we place 8 Q-toothpicks, so a(3) = 8+8 = 16.
At stage 4 we place 14 Q-toothpicks, so a(4) = 16+14 = 30.
After 4 stages in the Q-toothpick structure there are 1 circle, 2 "heads" and 12 exposed endpoints.

Nathaniel Johnston, Table of n, a(n) for n = 0..200
Nathaniel Johnston, C program for computing terms
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.]
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A160120, A160164, A187210, A187212, A187217, A187220.

a(15) - a(47) from Nathaniel Johnston, Apr 15 2011

A188156 If A187211 is written, starting at its fifth term, as a triangle with rows of lengths 2, 4, 8, 16, ..., the n-th row begins with the first 2^n-1 terms of the present sequence.

Original entry on oeis.org

22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 152, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032, 774, 280, 70, 120, 150, 168, 246, 360, 342, 232, 246, 376, 454, 568, 838, 1032

Offset: 1

Nathaniel Johnston, Mar 26 2011

nonn

Limiting behavior of the rows of the triangle in A187211.

			The triangle from A187211 begins:
22, 20
22, 40, 54, 40
22, 40, 54, 56, 70, 120, 134, 72
22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136
...
Thus this sequence is 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326...
The final entry of the n-th row (for n >= 2) is 16 + 8(2^n - 1).

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.]
Nathaniel Johnston, The Q-Toothpick Cellular Automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A147646, A187210, A187211.

a(35) corrected by Nathaniel Johnston at the suggestion of Omar E. Pol, Mar 28 2011

A282470 Q-toothpick sequence with Q-toothpicks of radius 1 and 2 (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 16, 40, 62, 102, 124, 204, 258, 338, 360, 440, 494, 606, 676, 916, 1050, 1194, 1216, 1296, 1350, 1462, 1532, 1772, 1906, 2082, 2152, 2392, 2542, 2878, 3124, 3844, 4170, 4442, 4464, 4544, 4598, 4710, 4780, 5020, 5154, 5330, 5400, 5640, 5790, 6126, 6372, 7092, 7418, 7722, 7792, 8032, 8182, 8518

Offset: 0

Omar E. Pol, Feb 16 2017

nonn

For the construction of this sequence we use the same the rules of A187210 (the Q-toothpick sequence) except that for the even-indexed generations here we use Q-toothpicks of radius 2, not 1.
The result is that the structure looks like an arrangement of ovals.
On the infinite square grid at stage 0 we start with no Q-toothpicks, so a(0) = 0.
For n >= 1, a(n) is the ratio between the total length of the lines of the structure after n-th stages and the length of a single Q-toothpick of radius 1.
A187210(n) gives the total number of Q-toothpicks in the structure after n-th stages.
A187211(n) gives the number of Q-toothpicks added at n-th stage.
Note that since the radius of the Q-toothpicks can be two distincts numbers so we can write an infinite number of sequences from cellular automata of this kind.

Cf. A282471 (essentially the first differences).
Cf. A187210 (Q-toothpick sequence).
Cf. A139250, A187212, A267694, A267698.

cp's OEIS Frontend

A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.

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A211022 Value on the axis "y" of the center of the n-th circle formed in the structure of A211000.

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A267694 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,1). The endpoints of the left hand Q-toothpick are at (0,1) and (1,2). The endpoints of the right hand Q-toothpick are at (1,0) and (2,1). With a(0) = 0.

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A267698 Q-toothpick sequence in the first quadrant starting with two Q-toothpicks centered at (1,3) and (3,1) respectively. The endpoints of the left hand Q-toothpick are at (0,3) and (1,4). The endpoints of the right hand Q-toothpick are at (3,0) and (4,1). With a(0) = 0.

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A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

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A267695 First differences of A267694.

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A267699 First differences of A267698.

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A187216 Q-toothpick sequence starting with two opposite Q-toothpicks centered at the same grid point.

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A188156 If A187211 is written, starting at its fifth term, as a triangle with rows of lengths 2, 4, 8, 16, ..., the n-th row begins with the first 2^n-1 terms of the present sequence.

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A282470 Q-toothpick sequence with Q-toothpicks of radius 1 and 2 (see Comments for precise definition).

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