A194270 -id:A194270 - cp's OEIS frontend

A220520 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind.

0, 1, 3, 7, 11, 15, 19, 27, 35, 39, 43, 51, 63, 79, 91, 107, 123, 127, 131, 139, 151, 167, 187, 211, 237, 261, 273, 293, 325, 365, 393, 425, 457, 461, 465, 473, 485, 501, 521, 545, 571, 595, 615, 647, 691, 755, 807, 855, 909, 944, 961, 981, 1017, 1065

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

The structure is essentially one of the vertical wedges of several D-toothpick structures. For more information see A220500. First differs from A194440 at a(14). The first differences (A220521) give the number of toothpicks or D-toothpicks added at n-th stage. See A220522 for the "narrow" triangle of the third kind.

Omar E. Pol, Illustration of the structure after 32 stages
Omar E. Pol, Illustration of the structure after 64 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A047999, A139250, A194270, A194440, A194442, A220494, A220521, A220522.

A194444 D-toothpick sequence of the second kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 106, 126, 151, 167, 171, 179, 195, 219, 247, 283, 325, 369, 389, 413, 453, 517, 549, 593, 646, 678, 682, 690, 706, 730, 758, 794, 838, 890, 932, 980, 1040, 1140, 1208, 1292, 1375, 1459, 1487, 1511, 1555

Offset: 0

Omar E. Pol, Aug 24 2011

nonn

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.

First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

Cf. A139250, A172310, A182838, A194270, A194432, A194434, A194440, A194442, A194445, A212008, A220524.

a(n) = A194434(n)/4. - Omar E. Pol, Oct 15 2011

More terms from Omar E. Pol, Mar 23 2013

A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 701, 749, 769, 777

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

The structure is essentially one of the oblique wedges of several D-toothpick structures. For more information see A220500. First differs from A194442 at a(47). The first differences (A220523) give the number of toothpicks or D-toothpicks added at n-th stage.

Omar E. Pol, Illustration of the structure after 32 stages
Omar E. Pol, Illustration of the structure after 64 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A047999, A139250, A194270, A194700, A194440, A194442, A194444, A220496, A220500, A220523.

A294020 Total number of elements after n-th stage of a hybrid cellular automaton formed by D-toothpicks and toothpicks (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 5, 9, 15, 23, 27, 41, 65, 81, 103, 111, 115, 129, 153, 169, 191, 199, 203, 217, 241, 257, 279, 287, 291, 305, 329, 345, 367, 375, 379, 393, 417, 433, 455, 463, 467, 481, 505, 521, 543, 551, 555, 569, 593, 609, 631, 639, 643, 657, 681, 697, 719, 727, 731, 745, 769, 785, 807, 815, 819, 833, 857, 873, 895, 903

Offset: 0

Omar E. Pol, Oct 21 2017

The sequence arises from a "hybrid" cellular automaton, which consist of two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.
The rules are the same as the rules of A290220 except that here the first element is only one D-toothpick, not two. The result is that here the structure has only two arms instead of four arms as in A290220. On the other hand the structure of each arm is more complex than the structure of the arms of A290220.
The behavior is similar to A289840 and A290220 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. For example: if here the first D-toothpick is placed in the NE-SW direction then the structure grows only in two opposite directions: NW and SE.
On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.
For the next stages we use the following rules:
1) If n is a number of the form 3*k + 1 (cf. A016777), for example: 1, 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected by their endpoints, in the same way as in the odd-indexed stages of A194270.
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), for example: 3, 6, 9, 12, 15, ..., 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.
The minimum width of the structure is 3*2^(1/2) = sqrt(18), see A010474.
The maximum width of the structure is 7*2^(1/2) = sqrt(98), see A010549.
The structure contains seven distinct polygons.
a(n) is the total number of elements in the structure after n generations.
A294021 (the first differences) gives the number of elements added at n-th stage.

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

Cf. A010474, A010549, A139250, A194270, A289840, A290220, A294021.

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)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Nov 12 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)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>10.
(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)

A319018 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition).

Original entry on oeis.org

0, 1, 9, 17, 57, 65, 121, 145, 265, 273, 329, 377, 617, 657, 865, 921, 1201, 1209, 1265, 1313, 1553, 1617, 2001, 2121, 2689, 2745, 3009, 3153, 3841, 3953, 4513, 4649, 5297, 5305, 5361, 5409, 5649, 5713, 6097, 6233, 6881, 6953, 7353, 7585, 8713, 8913, 9961

Offset: 0

Rémy Sigrist, Sep 08 2018

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON forever.
Each cell has 8 neighbors, the cells that are a knight's move away.
We begin in generation 1 with a single ON cell.
A cell is turned ON at generation n+1 if it has exactly one ON neighbor at generation n.
(Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has exactly one neighbor that has been turned ON at some earlier generation. - N. J. A. Sloane, Dec 19 2018)
This sequence has similarities with A151725: here we use knight moves, there we use king moves.
This is a knight's-move version of the Ulam-Warburton cellular automaton (see A147562). - N. J. A. Sloane, Dec 21 2018
The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections, which generate the dihedral group D_8 of order 8), so A319019 is a multiple of 8 (compare A322050). - N. J. A. Sloane, Dec 16 2018
From Omar E. Pol, Dec 16 2018: (Start)
For n >> 1 (for example: n = 257) the structure of this sequence is similar to the structure of both A194270 and of A220500, the D-toothpick cellular automata of the second kind and of the third kind respectively. The animations of both CAs are in the Applegate's movie version.
Also, the graph of A319018 is a bit similar to the graph of A245540, which is essentially a 45-degree-3D-wedge of A245542 (a pyramid) which is the partial sums of A160239 (Fredkin's replicator). See "Plot 2": A319018 vs. A245540. (End)
The conjecture that A322050(2^k+1)=1 also suggests a fractal geometry. Let P_k be the associated set of eight points. It appears that P_k may be written as the intersection of four fixed lines, y = +-2*x and x = +-2*y, with a circle, x^2 + y^2 = 5*4^k (see linked image "Log-Periodic Coloring"). - Bradley Klee, Dec 16 2018
In many of these toothpick or cellular automata sequences it is common to see graphs which look like some version of the famous blancmange curve (also known as the Takagi curve). I expect that is what we are seeing when we look at the graph of A322049, although we probably need to go a lot further out before the true shape becomes apparent. - N. J. A. Sloane, Dec 17 2018
The graph of A322049 (related to first differences of this sequence) appears to have rather a self-similar structure which repeats at powers of 2, and more specifically at 2^10 = 1024. There is no central symmetry or continuity, which are characteristic properties of the blancmange curve. - M. F. Hasler, Dec 28 2018
The 8 points added in generation n = 2^k + 1 are P_k = 2^k*K where K = {(+-2, +-1), (+-1, +-2)} is the set of the initial 8 knight moves. So P_k is indeed the intersection of the rays of slope +-1/2 resp. +-2 and a circle of radius 2^k*sqrt(5). In the subsequent generation n = 2^k + 2, the new cells switched on are exactly the 7 "new" knight move neighbors of these 8 cells, (P_k + K) \ (2^k - 1)*K. The 8th neighbor, lying one knight move closer to the origin, has been switched on in generation 2^k, together with an octagonal "wall" consisting of every other cell on horizontal and vertical segments between these points (2^k - 1)*K, and all cells on the diagonal segments between these points, as well as 2 more diagonals just next to these (on the inner side) and shorter by 2 cells (so they are empty for k = 1). This yields 4*(2 + (2^k - 2)*(1+3)) new ON cells in generation 2^k, plus 8*(2^(k-1) - 2) more new ON cells on horizontal, vertical and diagonal lines 4 units closer to the origin for k > 2, and similar additional terms for k > 4 etc. - M. F. Hasler, Dec 28 2018

Rémy Sigrist, Table of n, a(n) for n = 0..2049
David Applegate, The movie version.
Nathan Epstein, Gfycat animation of sequence.
M. F. Hasler, Interactive illustration of A319018 and A319019, Dec. 2018.
Bradley Klee, Log-Periodic Coloring at stage 257.
Bradley Klee, Log-periodic coloring of the first quadrant, over the chair tiling.
Rémy Sigrist, Illustration of the structure at stage 7
Rémy Sigrist, Illustration of the structure at stage 257
Rémy Sigrist, Colored illustration of the structure at stage 257 (where the hue is a function of the stage)
Rémy Sigrist, PARI program for A319018
N. J. A. Sloane, Hand-drawn sketch showing terms through about the eighth shell, but using offset a(0)=1. Illustrates the octagonal "castle walls".
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A151725, A319019 (first differences).
For further analysis see A322048, A322049, A322050, A322051.
See A322055, A322056 for a variation.
See also A139250, A147562, A194270, A220500.

A319018(n)=sum(i=1,n,A319019[i]) \\ with array A319019=A319019_upto(N) precomputed with sufficiently large N. - M. F. Hasler, Dec 28 2018

No formula or recurrence is presently known. See A322049 for a promising attack. - N. J. A. Sloane, Dec 16 2018

a(n) = Sum_{k=1..n} A319019(n) = 1 + 8*Sum_{k=2..n} A322050(n) for n >= 1. In particular, a(n) - 1 is divisible by 8 for all n >= 1. - M. F. Hasler, Dec 28 2018

Deleted an incorrect illustration. - N. J. A. Sloane, Dec 17 2018

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.

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

A194432 D-toothpick sequence starting with a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 12, 28, 44, 60, 88, 136, 168, 184, 216

Offset: 0

Omar E. Pol, Sep 03 2011

On the infinite square grid we start with no toothpicks.
At stage 1, we place a cross, centered at the origin, formed by 2 vertical toothpicks and 2 horizontal toothpicks of length 1. At stage 2, we place 8 D-toothpicks of length sqrt(2). At stage 3, we place 16 toothpicks of length 1. And so on.
The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194433) give the number of toothpicks and D-toothpicks added at n-th stage.
Apparently this cellular automaton has a fractal (or fractal-like) behavior related to power of 2, similar to A194270 and very similar to A194434. The octagonal structure contains a large number of distinct polygonal shapes. For more information see A194270, A194440 and A194442.

Cf. A139250, A194270, A194433, A194434, A194440, A194442.

A220524 D-toothpick sequence of the third kind in the first quadrant.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 34, 42, 46, 54, 70, 94, 110, 130, 155, 171, 175, 183, 199, 223, 255, 291, 333, 377, 397, 421, 465, 529, 569, 613, 666, 698, 702, 710, 726, 750, 782, 818, 862, 914, 956, 1004, 1068, 1168, 1248, 1332, 1423, 1507, 1535, 1559, 1607

Offset: 0

Omar E. Pol, Dec 15 2012

nonn

This is a toothpick sequence of forking paths to 135 degrees in the first quadrant. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. A220525 (the first differences) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure has a fractal (or fractal-like) behavior. For more information see A194700.

First differs from A194444 at a(13).

Cf. A139250, A194270, A194700, A194444, A220500, A220514, A220520, A220525.

a(n) = A220514(n)/4. - Omar E. Pol, Mar 23 2013

cp's OEIS Frontend

A220520 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "wide" triangle of the third kind.

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A194444 D-toothpick sequence of the second kind in the first quadrant.

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A220522 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the third kind.

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A294020 Total number of elements after n-th stage of a hybrid cellular automaton formed by D-toothpicks and toothpicks (see Comments lines for precise definition).

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

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A319018 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition).

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

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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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A194432 D-toothpick sequence starting with a cross formed by 4 toothpicks.

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