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

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

Author

Omar E. Pol, Jul 24 2017

Keywords

• nonn
• easy

Comments

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

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 (1,0,1,-1).

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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)