A187210 -id:A187210 - cp's OEIS frontend

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

1, 3, 5, 3, 4, 4, 5, 11, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17

Offset: 1

Omar E. Pol, Mar 24 2012

Consider a toothpick structure formed by L-toothpicks connected by their endpoints. The endpoints of the L-toothpicks are placed on the main diagonal of the first quadrant. At stage 1 we place an L-toothpick with one of its endpoints on the origin. At stage n we place an L-toothpick of size n. The L-toothpicks are placed alternately, on one or another sector of the first quadrant, trying to make the structure have an exposed endpoint closest to the origin. The total length of all L-toothpicks after the n-th stage is A002378(n). The value of x and y of the endpoint of the structure after the n-th stage is equal to the n-th term of Recamán's sequence A005132(n). Note that we can get other illustrations of initial terms of Recamán's sequence by replacing each L-toothpick by a Q-toothpick or by a semicircumference. This structure is also one of the three views of the three-dimensional model for Recamán's sequence. For more information about L-toothpicks and Q-toothpicks, see A172310 and A187210.

			The summands are the size of the L-toothpicks:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 2 + 3 = 5.
a(4) = 3.
a(5) = 4.
a(6) = 4.
a(7) = 5.
a(8) = 5 + 6 = 11.
a(9) = 6 + 7 = 13.
a(10) = 7.

First differences of A210607.

Cf. A005132, A064289, A119632, A139250, A160356, A160357, A171175, A171178, A172310, A187210, A210604, A210605, A210608-A210613.

A194434 D-toothpick sequence of the second kind starting with a X-shaped cross formed by 4 D-toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 92, 136, 168, 184, 216, 280, 376, 424, 504, 604, 668, 684, 716, 780, 876, 988, 1132, 1300, 1476, 1556, 1652, 1812, 2068, 2196, 2372, 2584, 2712, 2728, 2760, 2824, 2920, 3032, 3176, 3352, 3560, 3728, 3920, 4160, 4560, 4832, 5168

Offset: 0

Omar E. Pol, Sep 03 2011

nonn

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross as a "X", formed by 4 D-toothpicks of length sqrt(2) and centered at the origin. At stage 2, we place 8 toothpicks of length 1. At stage 3, we place 16 D-toothpicks of length sqrt(2). And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194435) give the number of toothpicks or D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal behavior (or fractal-like behavior) related to power of 2, similar to A194270 and very similar to A194432. The octagonal structure contains a large number of distinct closed polygonal regions. For more information see A194270, A194440 and A194442.
First differs from A220514 at a(13). - Omar E. Pol, Mar 23 2013
Observation: at least for the terms in the Data section the graph also appears to be a companion of the graph of A187210 but that could be different comparing more terms. - Omar E. Pol, Jun 24 2022

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animated version for n = 0..31
Paolo Xausa, Animated version for n = 0..63
Paolo Xausa, Illustration of initial terms for n = 0..63 (multipage PDF)
Index entries for sequences related to toothpick sequences

Cf. A139250, A187210, A194270, A194432, A194435, A194440, A194442, A194444, A212008, A220514.

a(n) = 4*A194444(n).

More terms from Omar E. Pol, Mar 23 2013

A187212 Q-toothpick sequence in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 21, 31, 39, 43, 51, 63, 75, 91, 119, 149, 165, 169, 177, 189, 201, 217, 245, 277, 297, 313, 341, 377, 417, 477, 565, 643, 675, 679, 687, 699, 711, 727, 755, 787, 807, 823, 851, 887, 927, 987, 1075

Offset: 0

Omar E. Pol, Mar 22 2011, Mar 30 2011

nonn

At stage 0, we start with no Q-toothpicks.
At stage 1, we place a Q-toothpick centered at (1,0) with its endpoints at (0,0) and (1,1).
At stage 2, we place two Q-toothpicks.
The sequence gives the number of Q-toothpicks in the structure after n-th stage.
For more information see A187210.
A187213 gives the number of Q-toothpicks added at n-th stage.
Note that starting from (0,1), with the first Q-toothpick centered at (1,1), we have the toothpick sequence A139250.
Also, gullwing sequence on the semi-infinite square grid, since a "gull" is formed by two Q-toothpicks. The sequence gives the number of "gulls" (or G-toothpicks) in the structure after n-th stage. See A187220. - Omar E. Pol, Mar 30 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..202
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, A160164, A187210, A187213, A187220.

It appears that a(n) = A139250(n) - 2*A059939(n), for n >= 1. - Omar E. Pol, Mar 29 2011

Terms after a(24) from Nathaniel Johnston, Mar 28 2011

A211001 Numbers congruent to 2 in the structure of A211000.

Original entry on oeis.org

2, 34, 42, 50, 150, 242, 246, 250, 354, 358, 370, 390, 394, 402, 406, 6570, 6602, 6606, 6622, 6626, 6630, 6634, 6654, 6658, 6682, 6686

Offset: 1

Omar E. Pol, Mar 30 2012

The behavior seems to be as modular arithmetic but in a growing structure. The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. For the number of circles at n-th stage see A211020.

Cf. A187210, A210838, A210841, A211000, A211002, A211003, A211010, A211011, A211020-A211023.

A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 0, 6, -4, 10, 1, 15, 7, 9, 14, 2, 22, 10, 13, 19, 3, 9, -8, -2, -20, 10, -7, 23, 7, 9, -8, -6, -24, -22, -7, -39, 11, -21, -8, -2, -28, -22, -7, -43, 15, -65, -8, -88, -32, -64, -7, -39, 19, -65, -8, -92, -36, -64, -65, -35, -95, -65, -64, -96

Offset: 0

Omar E. Pol, Mar 28 2012

It appears there is an infinite family of this type of curves or structures in which the terms of a sequence of positive integers are represented as inflection points and the gaps between them are essentially represented as nodes of spirals. For example: consider a structure formed by Q-toothpicks of size = Axxxxxa connected by their endpoints in which the inflection points are the exposed endpoints at stage Axxxxxb(n), where both Axxxxxa and Axxxxxb are sequences with positive integers. Also instead of Q-toothpicks we can use semicircumferences or also 3/4 of circumferences. For the definition of Q-toothpicks see A187210.
We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick of size 1 centered at (1,0) with its endpoints at (0,0) and (1,1). Since 1 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 2 we place a Q-toothpick of size 2 centered at (1,3) with its endpoints at (1,1) and (3,3).
At stage 3 we place a Q-toothpick of size 3 centered at (0,3) with its endpoints at (3,3) and (0,6). Since 3 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick of size 4 centered at (0,10) with its endpoints at (0,6) and (-4,10).
And so on...

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,       Yes
.    2         3,   3,        -
.    3         0,   6,       Yes
.    4        -4,  10,        -
.    5         1,  15,        -
.    6         7,   9,       Yes
.    7        14,   2,        -
.    8        22,  10,        -
.    9        13,  19,        -
.   10         3,   9,       Yes
.   11        -8,  -2,        -
.   12       -20,  10,        -
.   13        -7,  23,        -
.   14         7,   9,        -
.   15        -8,  -6,       Yes

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animation of terms n = 0..41 (first 21 coordinate pairs), where orange dots are toothpick endpoints (hollow dots are inflection points) and blue dots are toothpick centers
Paolo Xausa, Animation of terms n = 0..507 (first 254 coordinate pairs)
Paolo Xausa, Scatterplot of 10^5 coordinate pairs
Index entries for sequences related to toothpick sequences

Cf. A210841 (the same idea for primes).

Cf. A000217, A005132, A187210, A210606, A211000, A211011.

A210838[nmax_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2, infl=0}, Join[{ep}, Table[If[n>1&&IntegerQ[Sqrt[8(n-1)+1]], infl++, If[Mod[infl, 2]==1, turn*=-1]; angle-=turn; infl=0]; ep=AngleVector[ep, {Sqrt[2]n, angle}], {n, nmax}]]];
A210838[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Jan 12 2023 *)

A210838(nmax) = my(ep=vector(nmax+1), turn=1, infl=0, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(issquare((n-1)<<3+1), infl++; ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(infl%2, turn*=-1); infl=0; ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210838(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

from numpy import sign
from sympy import integer_nthroot
def A210838(nmax):
    ep, turn, infl = [(0, 0), (1, 1)], 1, 0
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if integer_nthroot(((n - 1) << 3) + 1, 2)[1]: # Continue straight
            infl += 1
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if infl % 2: turn *= -1
            infl = 0
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210838(100)) # Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

a(30)-a(33) corrected and more terms by Paolo Xausa, Jan 12 2023

A210841 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is prime.

Original entry on oeis.org

0, 0, 1, 1, 3, -1, 6, -4, 10, -8, 5, -13, -1, -19, 6, -26, 14, -34, 5, -43, -5, -33, 6, -22, 18, -10, 5, 3, -9, 17, 6, 32, 22, 16, 5, -1, -13, -19, 6, -38, 26, -58, 5, -79, -17, -57, 6, -34, 30, -10, 5, 15, -21, -11, 6, -38, 34, -10, 5, 19, -25, 49, 6, 80, 38, 112

Offset: 0

Omar E. Pol, Mar 29 2012

The same idea as A210838 but here the inflection points are prime numbers.

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,        -
.    2         3,  -1,       Yes
.    3         6,  -4,       Yes
.    4        10,  -8,        -
.    5         5, -13,       Yes
.    6        -1, -19,        -
.    7         6, -26,       Yes

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animation of terms n = 0..41 (first 21 coordinate pairs), where orange dots are toothpick endpoints (hollow dots are inflection points) and blue dots are toothpick centers
Index entries for sequences related to toothpick sequences

Cf. A000040, A187210, A210606, A210838, A211000, A211011.

A210841[nmax_]:=Module[{ep={0,0},angle=3/4Pi,turn=Pi/2},Join[{ep},Table[If[!PrimeQ[n-1],If[n>6&&PrimeQ[n-2],turn*=-1];angle-=turn];ep=AngleVector[ep,{Sqrt[2]n,angle}],{n,nmax}]]];
A210841[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Mar 04 2023 *)

A210841(nmax) = my(ep=vector(nmax+1), turn=1, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(isprime(n-1), ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(n>6 && isprime(n-2), turn*=-1); ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210841(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Mar 04 2023

from numpy import sign
from sympy import isprime
def A210841(nmax):
    ep, turn = [(0, 0), (1, 1)], 1
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if isprime(n - 1): # Continue straight
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if n > 6 and isprime(n - 2): turn *= -1
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210841(100)) # Generates 101 coordinate pairs - Paolo Xausa, Mar 04 2023

a(14) corrected by and more terms from Paolo Xausa, Mar 04 2023

A211023 Value on the axis "y" of the endpoint of the structure of A211000 if the index is prime.

Original entry on oeis.org

0, -1, -3, -5, -5, -3, -3, -5, -5, -3, -1, 1, 1, -1, -1, 1, 3, 5, 7, 7, 5, 3, 3, 5, 5, 5, 7, 7, 5, 5, 7, 7, 5, 3, 1, -1, -3, -5, -5, -3, -1, 1, 3, 5, 5, 3, 3, 3, 3, 5, 5, 3, 1, -1, -3, -5, -7, -9, -11, -11, -9, -7, -5, -5, -7, -7, -5, -3, -1, 1, 1, -1, -1, 1, 3

Offset: 1

Omar E. Pol, Mar 31 2012

sign

a(n) is also the value on the axis "y" of the n-th inflection point in the structure of A211000.

The behavior seems to be as modular arithmetic but in a growing structure. The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure of A211000 arises from the prime numbers A000040.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A187210, A211000, A211001, A211002, A211003, A211010, A211011, A211020, A211021, A211022, A211024.

A211023[upto_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2}, Table[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; ep=AngleVector[ep, {Sqrt[2], angle}];If[PrimeQ[n+1], Last[ep], Nothing], {n, 0,upto-1}]];
A211023[500] (* Paolo Xausa, Jan 14 2023 *)

a(n) = A211011(A000040(n)).

A210607 Vertex number of an L-toothpick structure which give Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 20, 25, 36

Offset: 1

Omar E. Pol, Mar 24 2012

For more information see A210606.

First differences give A210606.

A002378, A005132, A119632, A139250, A160356, A171175, A171178, A172310, A187210, A210607-A210613.

A211008 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after n-th stage in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 4, 4, 4, 8, 8, 2, 8, 12, 4, 8, 12, 4, 12, 12, 4, 16, 16, 4, 16, 20, 4, 20, 20, 4, 32, 28, 4, 40, 44, 8, 2, 40, 52, 12, 4, 40, 52, 12, 4, 44, 52, 12, 4, 48, 56, 12, 4, 48, 60, 12, 4, 52, 60, 12, 4, 64, 68, 12, 4, 72, 84, 16, 4

Offset: 1

Omar E. Pol, Sep 18 2012

It appears that the number of rectangles of area 2 in the toothpick structure of A139250 equals the number of hearts in the Q-toothpick cellular automaton of A187210. See conjecture in formula section.

			For n = 8 in the toothpick structure after 8 stages we have that:
T(8,1) = 8 is the number of squares of size 1 X 1.
T(8,2) = 12 is the number of rectangles of size 1 X 2.
T(8,3) = 4 is the number of squares of size 2 X 2.
Written as an irregular array the sequence begins:
   0;
   0;
   0,  2;
   0,  4;
   0,  4;
   4,  4;
   8,  8,  2;
   8, 12,  4;
   8, 12,  4;
  12, 12,  4;
  16, 16,  4;
  16, 20,  4;
  20, 20,  4;
  32, 28,  4;
  40, 44,  8,  2;
  40, 52, 12,  4;

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Zero together with the row sums gives A160124.

Cf. A139250, A159786, A160125, A168131, A187210, A188346, A211016, A211017, A211019.

It appears that T(n,2) = A188346(n+2) (checked by hand up to n = 128 in the toothpick structure of A139250).

A211021 Numbers n such that a new circle appears in the structure of A211000.

Original entry on oeis.org

11, 13, 15, 34, 41, 65, 71, 75, 275, 281, 285, 437, 443, 561, 567, 575, 581, 591, 597, 605, 611, 617, 647, 663, 957, 971, 1025, 1037, 1043, 1055, 1067, 1073, 1091, 1113, 1153, 1165, 1711, 2243, 3377, 3467, 5809, 7937, 7955, 8021, 8043, 8057, 8063

Offset: 1

Omar E. Pol, Mar 31 2012

nonn

Also where the positive records occur in A211020.

The behavior seems to be as modular arithmetic but in a growing structure. The structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040.

Paolo Xausa, Table of n, a(n) for n = 1..250
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A187210, A211000-A211003, A211010, A211011, A211020, A211022-A211024.

A211020[nmax_]:=Module[{ep={{0, 0}}, angle=3/4Pi, turn=Pi/2, cells}, Join[{0}, Table[If[!PrimeQ[n], If[n>5&&PrimeQ[n-1], turn*=-1]; angle-=turn]; AppendTo[ep, AngleVector[Last[ep], {Sqrt[2], angle}]]; cells=FindCycle[Graph[MapApply[UndirectedEdge, Partition[ep, 2, 1]]], {4}, All]; CountDistinct[Map[Sort, Map[First, cells, {2}]]], {n, 0, nmax-1}]]];
Flatten[Position[Differences[A211020[1000]],1]] (* Paolo Xausa, Jan 16 2023 *)

cp's OEIS Frontend

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

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A194434 D-toothpick sequence of the second kind starting with a X-shaped cross formed by 4 D-toothpicks.

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A187212 Q-toothpick sequence in the first quadrant.

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A211001 Numbers congruent to 2 in the structure of A211000.

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A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

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A210841 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is prime.

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A211023 Value on the axis "y" of the endpoint of the structure of A211000 if the index is prime.

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A210607 Vertex number of an L-toothpick structure which give Recamán's sequence A005132.

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A211008 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after n-th stage in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

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