

A289840


Complex cross sequence (see Comments lines for definition).


8



0, 1, 3, 11, 19, 27, 35, 67, 83, 99, 115, 163, 179, 195, 211, 275, 323, 355, 387, 467, 483, 499, 515, 579, 627, 675, 707, 787, 803, 819, 835, 899, 947, 995, 1027, 1107, 1123, 1139, 1155, 1219, 1267, 1315, 1347, 1427, 1443, 1459, 1475, 1539, 1587, 1635, 1667, 1747, 1763, 1779, 1795, 1859, 1907, 1955, 1987, 2067
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OFFSET

0,3


COMMENTS

The sequence arises from a "hybrid" cellular automaton on the infinite square grid, which consist in two successive generations using toothpicks of length 2 (cf. A139250) followed by two successive generations using the rules of the Dtoothpick sequence A220500.
In other words (and more precisely) we have that:
1) If n is congruent to 1 or 2 mod 4 (cf. A042963), for example: 1, 2, 5, 6, 9, 10, ..., the elements added to the structure at nth stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
2) If n is a positive integer of the form 4*k1 (cf. A004767), for example: 3, 7, 11, 15, ..., the elements added to the structure at nth stage must be Dtoothpicks of length sqrt(2) and eventually Dtoothpicks of length sqrt(2)/2, in both cases the Dtoothpicks are connected to the structure by their endpoints, in the same way as in the evenindexed stages of A220500.
3) If n is a positive multiple of 4 (cf. A008586) the elements added to the structure at nth stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the oddindexed stages of A220500.
a(n) is the total number of elements in the structure after n generations.
A289841 (the first differences) gives the number of elements added at nth stage.
Note that after 19 generations the structure is a 72gon which essentially looks like a diamond (as a square that has been rotated 45 degrees).
The surprising fact is that from n = 20 up to 27 the structure is gradually transformed into a square cross.
The diamond mentioned above can be interpreted as the center of the cross. The diamond has an area equal to 384 and it contains 222 polygonal regions (or enclosures) of 11 distinct shapes. Missing two heptagonal shapes which are in the arms of the square cross only.
In total the complex cross contains 13 distinct shapes of polygonal regions. There are ten polygonal shapes that have an infinite number of copies. On the other hand, three of these polygonal shapes have a finite number of copies because they are in the center of the cross only. For example: there are only four copies of the concave 14gon, which is also the largest polygon in the structure.
For n => 27 the shape of the square cross remains forever because its four arms grow indefinitely.
Every arm has a minimum width equal to 8, and a maximum width equal to 12.
Every arm also has a periodic structure which can be dissected in infinitely many clusters of area equal to 64. Every cluster is a 30gon that contains 40 polygonal regions of nine distinct shapes.
If n is a number of the form 8*k3 (cf. A017101) and greater than 19, for example: 27, 35, 43, 51, ..., then at nth stage a new cluster is finished in every arm of the cross.
The behavior is similar to A290220 and A294020 in the sense that these three cellular automata have the property of selflimiting 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 (1,0,0,0,0,0,0,1,1).


FORMULA

From Colin Barker, Nov 11 2017: (Start)
G.f.: x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1  x)^2*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n1) + a(n8)  a(n9) for n>19.
(End)


PROG

(PARI) concat(0, Vec(x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1  x)^2*(1 + x)*(1 + x^2)*(1 + x^4))+ O(x^50))) \\ Colin Barker, Nov 12 2017


CROSSREFS

Cf. A004767, A008586, A017101, A042963, A139250, A220500, A289841, A290220 (a simpler cross), A294020.
Sequence in context: A078583 A017101 A112422 * A125994 A137295 A038536
Adjacent sequences: A289837 A289838 A289839 * A289841 A289842 A289843


KEYWORD

nonn,easy


AUTHOR

Omar E. Pol, Jul 14 2017


STATUS

approved



