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

%I A290220 #52 Aug 27 2024 16:51:45
%S A290220 0,2,6,10,18,26,34,42,58,70,78,94,106,114,130,142,150,166,178,186,202,
%T A290220 214,222,238,250,258,274,286,294,310,322,330,346,358,366,382,394,402,
%U A290220 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
%N A290220 Narrow cross sequence (see Comments lines for definition).
%C A290220 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.
%C A290220 On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.
%C A290220 For the next stages we have the following rules:
%C A290220 1) At stage 1 we place two D-toothpicks connected by their endpoints on the same diagonal.
%C A290220 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.
%C A290220 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.
%C A290220 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.
%C A290220 a(n) is the total number of elements in the structure after n generations.
%C A290220 A290221 (the first differences) gives the number of elements added at n-th stage.
%C A290220 The surprising fact is that from n = 7 up to 9 the structure is gradually transformed into a square cross.
%C A290220 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.
%C A290220 Every arm has a width equal to 4.
%C A290220 Every arm also has a periodic structure which can be dissected in infinitely many clusters.
%C A290220 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.
%C A290220 The structure looks like a square cross but it's simpler than the structure of the complex cross described in A289840.
%C A290220 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
%H A290220 Colin Barker, <a href="/A290220/b290220.txt">Table of n, a(n) for n = 0..1000</a>
%H A290220 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A290220 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290220 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A290220 From _Colin Barker_, Nov 11 2017: (Start)
%F A290220 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)).
%F A290220 a(n) = a(n-1) + a(n-3) - a(n-4) for n>9. [Corrected by _Paolo Xausa_, Aug 27 2024]
%F A290220 (End)
%t A290220 LinearRecurrence[{1, 0, 1, -1}, {0, 2, 6, 10, 18, 26, 34, 42, 58, 70}, 100] (* _Paolo Xausa_, Aug 27 2024 *)
%o A290220 (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
%Y A290220 Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289840, A290221, A294020.
%K A290220 nonn,easy
%O A290220 0,2
%A A290220 _Omar E. Pol_, Jul 24 2017