A289841 Number of elements added at n-th stage to the structure of the complex square cross described in A289840.

0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48

Offset: 0

Author

Omar E. Pol, Jul 14 2017

Keywords

• nonn
• tabf
• easy

Comments

For n = 0..17 the sequence is similar to the known toothpick sequences.

The surprising fact is that for n >= 18 the sequence has a periodic tail. More precisely, it has period 8: repeat [32, 80, 16, 16, 16, 64, 48, 48]. This tail is in accordance with the expansion of the four arms of the cross. The tail also can be written starting from the 20th stage, with period 8: repeat [16, 16, 16, 64, 48, 48, 32, 80], (see example).

This sequence is essentially the first differences of A289840. The behavior is similar to A290221 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

Examples

			For n = 0..17 the sequence is 0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32;
Terms 18 and beyond can be arranged in a rectangular array with eight columns as shown below:
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
...
On the other hand, in accordance with the periodic structure of the arms of the square cross, the terms 20 and beyond can be arranged in a rectangular array with eight columns as shown below:
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
...
		

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 (0,0,0,0,0,0,0,1).

Crossrefs

Cf. A004767, A008584, A042963, A139250, A139251, A220500, A220501, A289840, A290221, A294021.

Programs

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)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^100))) \\ Colin Barker, Nov 12 2017

Formula

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)*(1 + x)*(1 + x^2)*(1 + x^4)). - Colin Barker, Nov 12 2017