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).

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

Author

Omar E. Pol, Oct 21 2017

Keywords

• nonn
• easy

Comments

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.

Links

• 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).

Crossrefs

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

Programs

PARI

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

Formula

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)