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

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

Author

Omar E. Pol, Jan 21 2019

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

• 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
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

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.

Programs

Mathematica

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 *)

Formula

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.