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A160414
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Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).
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15
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0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745
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OFFSET
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0,3
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COMMENTS
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The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For an illustration of 1729 (the Hardy-Ramanujan number) see link section.
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LINKS
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Table of n, a(n) for n=0..46.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
O. E. Pol, Illustration of initial terms
O. E. Pol, Illustration of the structure after 24th stage (there are 1729 ON cells)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
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FORMULA
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a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012
a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013
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EXAMPLE
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If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
99.............99
98888888888888889
.877887787788778.
.876666787666678.
.886556888655688.
.886544444445688.
.876643343346678.
.877843222348778.
.888844212448888.
.877843222348778.
.876643343346678.
.886544444445688.
.886556888655688.
.876666787666678.
.877887787788778.
98888888888888889
99.............99
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2"'s around the central cell, leading to a(2)=a(1)+8=9. In the third generation, twelve "3"'s are turned ON around the vertices of the square, a(3)=a(2)+3*4=21, and so on.
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MAPLE
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Contribution from R. J. Mathar, Oct 16 2010: (Start)
read("transforms") ; isA000079 := proc(n) if type(n, 'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc:
A160414 := proc(n) add( A161415(k), k=1..n) ; end proc: seq(A160414(n), n=0..90) ; (End)
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CROSSREFS
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Cf. A139250, A160118, A160410, A160412.
Cf. A001235, A011541, A000225, A060867, A160117, A161415, A160720, A160727, A151725.
Sequence in context: A135187 A173391 A133762 * A118130 A144482 A176896
Adjacent sequences: A160411 A160412 A160413 * A160415 A160416 A160417
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KEYWORD
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nonn
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AUTHOR
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Omar E. Pol, May 20 2009, Jun 13 2009, Jun 14 2009
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EXTENSIONS
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Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009
More terms from R. J. Mathar, Oct 16 2010
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STATUS
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approved
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