A294020 -id:A294020 - cp's OEIS frontend

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

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)

A010474 Decimal expansion of square root of 18.

Original entry on oeis.org

4, 2, 4, 2, 6, 4, 0, 6, 8, 7, 1, 1, 9, 2, 8, 5, 1, 4, 6, 4, 0, 5, 0, 6, 6, 1, 7, 2, 6, 2, 9, 0, 9, 4, 2, 3, 5, 7, 0, 9, 0, 1, 5, 6, 2, 6, 1, 3, 0, 8, 4, 4, 2, 1, 9, 5, 3, 0, 0, 3, 9, 2, 1, 3, 9, 7, 2, 1, 9, 7, 4, 3, 5, 3, 8, 6, 3, 2, 1, 1, 1, 6, 5, 5, 1, 1, 6, 2, 6, 0, 2, 9, 8, 2, 9, 2, 4, 7, 1

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {4, 8} repeated. - Harry J. Smith, Jun 05 2009
3*sqrt(2) = sqrt(18) is also the minimum width of the arms of the cellular automaton described in A294020. - Omar E. Pol, Oct 29 2017
Equals lambda(3) where lambda(n) = n*Product_{k=2..n-1} k^((n-k)/(n-k+1)). - Michel Marcus, Apr 02 2020

			4.242640687119285146405066172629094235709015626130844219530039213972197....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Michael Penn, A nice algebra problem with complex numbers, YouTube video, 2022.
Index entries for algebraic numbers, degree 2

Cf. A040013 (continued fraction). - Harry J. Smith, Jun 03 2009

SetDefaultRealField(RealField(100)); Sqrt(18); // G. C. Greubel, Feb 14 2019

evalf[100](sqrt(18)); # Muniru A Asiru, Feb 12 2019

RealDigits[Sqrt[18], 10, 100][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 100); sqrt(18) \\ Harry J. Smith, Jun 03 2009 (modified by G. C. Greubel, Feb 14 2019)

numerical_approx(sqrt(18), digits=100) # G. C. Greubel, Feb 14 2019

A289840 Complex cross sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 11, 19, 27, 35, 67, 83, 99, 115, 163, 179, 195, 211, 275, 323, 355, 387, 467, 483, 499, 515, 579, 627, 675, 707, 787, 803, 819, 835, 899, 947, 995, 1027, 1107, 1123, 1139, 1155, 1219, 1267, 1315, 1347, 1427, 1443, 1459, 1475, 1539, 1587, 1635, 1667, 1747, 1763, 1779, 1795, 1859, 1907, 1955, 1987, 2067

Offset: 0

Omar E. Pol, Jul 14 2017

The sequence arises from a "hybrid" cellular automaton on the infinite square grid, which consist of two successive generations using toothpicks of length 2 (cf. A139250) followed by two successive generations using the rules of the D-toothpick sequence A220500.
In other words (and more precisely) we have that:
1) If n is congruent to 1 or 2 mod 4 (cf. A042963), for example: 1, 2, 5, 6, 9, 10, ..., the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
2) If n is a positive integer of the form 4*k-1 (cf. A004767), for example: 3, 7, 11, 15, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) and eventually D-toothpicks of length sqrt(2)/2, in both cases the D-toothpicks are connected to the structure by their endpoints, in the same way as in the even-indexed stages of A220500.
3) If n is a positive multiple of 4 (cf. A008586) the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the odd-indexed stages of A220500.
a(n) is the total number of elements in the structure after n generations.
A289841 (the first differences) gives the number of elements added at n-th stage.
Note that after 19 generations the structure is a 72-gon which essentially looks like a diamond (as a square that has been rotated 45 degrees).
The surprising fact is that from n = 20 up to 27 the structure is gradually transformed into a square cross.
The diamond mentioned above can be interpreted as the center of the cross. The diamond has an area equal to 384 and it contains 222 polygonal regions (or enclosures) of 11 distinct shapes. Missing two heptagonal shapes which are in the arms of the square cross only.
In total the complex cross contains 13 distinct shapes of polygonal regions. There are ten polygonal shapes that have an infinite number of copies. On the other hand, three of these polygonal shapes have a finite number of copies because they are in the center of the cross only. For example: there are only four copies of the concave 14-gon, which is also the largest polygon in the structure.
For n => 27 the shape of the square cross remains forever because its four arms grow indefinitely.
Every arm has a minimum width equal to 8, and a maximum width equal to 12.
Every arm also has a periodic structure which can be dissected in infinitely many clusters of area equal to 64. Every cluster is a 30-gon that contains 40 polygonal regions of nine distinct shapes.
If n is a number of the form 8*k-3 (cf. A017101) and greater than 19, for example: 27, 35, 43, 51, ..., then at n-th stage a new cluster is finished in every arm of the cross.
The behavior is similar to A290220 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

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 (1,0,0,0,0,0,0,1,-1).

Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289841, A290220 (a simpler cross), A294020.

concat(0, Vec(x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4))+ O(x^50))) \\ Colin Barker, Nov 12 2017

From Colin Barker, Nov 11 2017: (Start)
G.f.: x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n-1) + a(n-8) - a(n-9) for n>19.
(End)

A290220 Narrow cross sequence (see Comments lines for definition).

Original entry on oeis.org

0, 2, 6, 10, 18, 26, 34, 42, 58, 70, 78, 94, 106, 114, 130, 142, 150, 166, 178, 186, 202, 214, 222, 238, 250, 258, 274, 286, 294, 310, 322, 330, 346, 358, 366, 382, 394, 402, 418, 430, 438, 454, 466, 474, 490, 502, 510, 526, 538, 546, 562, 574, 582, 598, 610, 618, 634, 646, 654, 670, 682, 690, 706, 718, 726, 742, 754

Offset: 0

Omar E. Pol, Jul 24 2017

The sequence arises from a "hybrid" cellular automaton, which consist essentially in two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.
On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.
For the next stages we have the following rules:
1) At stage 1 we place two D-toothpicks connected by their endpoints on the same diagonal.
2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.
3) If n is a positive multiple of 3 (cf. A008585) the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
4) If n is a number of the form 3*k + 1 (cf. A016777) and > 1, for example: 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected to the structure by their endpoints, in the same way as in the odd-indexed stages of A194270.
a(n) is the total number of elements in the structure after n generations.
A290221 (the first differences) gives the number of elements added at n-th stage.
The surprising fact is that from n = 7 up to 9 the structure is gradually transformed into a square cross.
For n => 9 the shape of the square cross remains forever because its four arms grow indefinitely in the directions North, East, West and South.
Every arm has a width equal to 4.
Every arm also has a periodic structure which can be dissected in infinitely many clusters.
In total, the narrow cross contains five distinct shapes of polygonal regions. There are three polygonal shapes that have an infinite number of copies. On the other hand, two polygonal shapes have a finite number of copies because they are in the center of the cross only. they are the heptagon and the hexagon of area 5.
The structure looks like a square cross but it's simpler than the structure of the complex cross described in A289840.
The behavior is similar to A289840 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

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 (1,0,1,-1).

Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289840, A290221, A294020.

LinearRecurrence[{1, 0, 1, -1}, {0, 2, 6, 10, 18, 26, 34, 42, 58, 70}, 100] (* Paolo Xausa, Aug 27 2024 *)

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)^2*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Nov 12 2017

From Colin Barker, Nov 11 2017: (Start)
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)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>9. [Corrected by Paolo Xausa, Aug 27 2024]
(End)

A323650 Flower garden sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 15, 19, 27, 39, 63, 67, 75, 87, 111, 123, 147, 183, 255, 259, 267, 279, 303, 315, 339, 375, 447, 459, 483, 519, 591, 627, 699, 807, 1023, 1027, 1035, 1047, 1071, 1083, 1107, 1143, 1215, 1227, 1251, 1287, 1359, 1395, 1467, 1575, 1791, 1803, 1827, 1863, 1935, 1971, 2043, 2151, 2367, 2403, 2475

Offset: 0

Omar E. Pol, Jan 21 2019

nonn

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a V-toothpick formed of two single toothpicks such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.
If n is odd then we add I-toothpicks in vertical position (see the example).
a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
A323651 (the first differences) gives the number of elements added at the n-th stage.
Note that 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains only three kinds of polygonal regions as follows:
- Rhombuses that contain two triangular cells.
- Regular hexagons that contain six triangular cells.
- Oblong hexagons that contain 10 triangular cells.
The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.
Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:
1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.
2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).
3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.
4) For more connections to A147562 see the Formula section.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - Omar E. Pol, Mar 06 2019

			Illustration of initial terms:
.
                        |   |
                \ /     |\ /|
         |       |        |
         |       |        |
                / \     |/ \|
                        |   |
n        1       2        3
a(n)     1       3        7
.
Note that for n = 2 the structure is also the same as a pair of Y-toothpicks connected by their endpoints (see A160120).

Cf. A002450, A103454, A139250 (normal toothpicks), A147562 (Ulam-Warburton), A147582, A160120 (Y-toothpicks), A161206 (V-toothpicks), A296612, A323641, A323642, A323649 (corner sequence), A323651 (first differences).

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770.

bw3[n_]:=bw3[n]=3^(DigitCount[n,2,1]-1);
A147562[n_]:=A147562[n]=If[n<2,n,1+4Sum[bw3[k],{k,n-1}]];
A323650[n_]:=If[OddQ[n],3A147562[(n-1)/2]+A147562[n]-A147562[n-1],3A147562[n/2]];
nterms=100;Array[A323650,nterms,0] (* Paolo Xausa, Jun 17 2022 *)

a(n) = 3*A147562(n/2) if n is even.
a(n) = 3*A147562((n-1)/2) + A147582(n) if n is odd.
a(n) = 3*A147562((n-1)/2) + A147562(n) - A147562(n-1) if n is odd.
a(2^n) = A103454(n), n >= 0.

A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 5, 13, 21, 33, 53, 69, 77, 101, 117

Offset: 0

Omar E. Pol, Nov 12 2017

It seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints in the structure, so the structure is finished.
A294963(n) gives the number of elements added to the structure at n-th stage.

Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500, A294963.

For other hybrid cellular automata, see A289840, A290220, A294020, A294980.

A294980 a(n) is the total number of elements after n-th stage in a hybrid cellular automaton formed by Y-toothpicks and V-toothpicks (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 40, 58, 76, 82

Offset: 0

Omar E. Pol, Nov 12 2017

We are on the infinite triangular grid.
At stage 0 there are no elements in the structure, so a(0) = 0.
If n is odd at n-th stage we add Y-toothpicks to the structure.
If n is a positive even number at n-th stage we add V-toothpicks to the structure.
a(n) is the total number of Y-toothpicks and V-toothpicks after n-th stages.
A294981(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "ab". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Cf. A139250, A160120 (Y-toothpicks), A161206 (V-toothpicks), A294981 (first differences), A296612.

For other hybrid cellular automata see: A289840, A290220, A294020, A294962.

A010549 Decimal expansion of square root of 98.

Original entry on oeis.org

9, 8, 9, 9, 4, 9, 4, 9, 3, 6, 6, 1, 1, 6, 6, 5, 3, 4, 1, 6, 1, 1, 8, 2, 1, 0, 6, 9, 4, 6, 7, 8, 8, 6, 5, 4, 9, 9, 8, 7, 7, 0, 3, 1, 2, 7, 6, 3, 8, 6, 3, 6, 5, 1, 2, 2, 3, 6, 7, 5, 8, 1, 6, 5, 9, 3, 5, 1, 2, 7, 3, 4, 9, 2, 3, 4, 7, 4, 9, 2, 7, 1, 9, 5, 2, 7, 1, 2, 7, 4, 0, 2, 9, 3, 4, 9, 1, 0, 0, 9, 1, 4, 5, 0, 9

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {1, 8, 1, 18} repeated. - Harry J. Smith, Jun 12 2009

7*2^(1/2) = sqrt(98) is also the maximum width of the arms of the cellular automaton described in A294020. - Omar E. Pol, Oct 29 2017

			9.899494936611665341611821069467886549987703127638636512236758165935127...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A010169 Continued fraction.

RealDigits[N[Sqrt[98],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2012 *)
RealDigits[Sqrt[98],10,120][[1]] (* Harvey P. Dale, May 21 2023 *)

{ default(realprecision, 20080); x=sqrt(98); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010549.txt", n, " ", d)); } \\ Harry J. Smith, Jun 12 2009

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A299770 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton on the infinite square grid, formed by toothpicks of length 2, D-toothpicks, toothpicks of length 1, and T-toothpicks.

Original entry on oeis.org

1, 5, 13, 21, 33, 49, 65, 73, 97, 105

Offset: 1

Omar E. Pol, Mar 20 2018

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).
The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.
After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.
A299771(n) gives the number of elements added to the structure at n-th stage.
The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

Very similar to A294962.
Cf. A299771, A296612.
Cf. A139250 (toothpicks), A160172 (T-toothpicks), A194700 (D-toothpicks), A220500.
For other hybrid cellular automata, see A194270, A220500, A289840, A290220, A294020, A294980.

A323646 "Letter A" toothpick sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 21, 27, 39, 53, 65, 71, 83, 97, 113, 131, 163, 197, 217, 223, 235, 249, 265, 283, 315, 349, 373, 391, 423, 461, 505, 567, 659, 741, 777, 783, 795, 809, 825, 843, 875, 909, 933, 951, 983, 1021, 1065, 1127, 1219, 1301, 1341, 1359, 1391, 1429, 1473, 1535, 1627, 1713, 1773, 1835, 1931

Offset: 0

Omar E. Pol, Mar 07 2019

nonn

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).
For the construction of the sequence the rules are as follows:
On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.
For the next n generations we have that:
At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.
If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.
If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.
If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.
The sequence gives the number of toothpicks and D-toothpicks after n stages.
A323647 (the first differences) gives the number of elements added at the n-th stage.
Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

			After two generations the structure looks like a letter "A" which is formed by a initial I-toothpick (or a toothpick of length 2), placed in horizontal direction, and two D-toothpicks each of length 2*sqrt(2) as shown below, so a(2) = 3.
Note that angle between both D-toothpicks is 90 degrees.
.
                      *
                    *   *
                  * * * * *
                *           *
              *               *
.
After three generations the structure contains three horizontal toothpicks and two D-toothpicks as shown below, so a(3) = 5.
.
                      *
                    *   *
                  * * * * *
                *           *
          * * * * *       * * * * *
.

Cf. A139250, A168112, A160730, A296612, A323647.

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A292612, A299770, A323650, A327330, A327332.

a(n) = 1 + A160730(n-1), n >= 1.

a(n) = 1 + 2*A168112(n-1), n >= 1.

cp's OEIS Frontend

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

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A010474 Decimal expansion of square root of 18.

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Magma

Maple

Mathematica

PARI

Sage

A289840 Complex cross sequence (see Comments lines for definition).

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PARI

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A290220 Narrow cross sequence (see Comments lines for definition).

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Mathematica

PARI

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A323650 Flower garden sequence (see Comments for precise definition).

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Mathematica

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A294962 a(n) is the total number of elements after n-th stage of a hybrid (and finite) cellular automaton formed by toothpicks, D-toothpicks and T-toothpicks on the infinite square grid (see Comments lines for precise definition).

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A294980 a(n) is the total number of elements after n-th stage in a hybrid cellular automaton formed by Y-toothpicks and V-toothpicks (see Comments lines for precise definition).

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A010549 Decimal expansion of square root of 98.

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