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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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19
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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 the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.
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LINKS
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FORMULA
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EXAMPLE
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With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21, 49;
61, 97, 133, 225;
237, 273, 309, 417, 453, 561, 669, 961;
...
This triangle T(n,k) shares with the triangle A256530 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
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Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_|_ | |_ _| | |
. | |_ _| | |_| _ |_| | |_ _| |
. | _ _ | _| |_| |_ | _ _ |
. | | _|_| | |_ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
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After 10 generations there are 273 ON cells, so a(10) = 273.
(End)
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MAPLE
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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:
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MATHEMATICA
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A160414list[nmax_]:=Accumulate[Table[If[n<2, n, 4*3^DigitCount[n-1, 2, 1]-If[IntegerQ[Log2[n]], 2n, 0]], {n, 0, nmax}]]; A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)
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PROG
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(PARI) my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<<valuation(n, 2)), n\2)); vector(99, i, 4*(s+=t(i))+1) \\ Altug Alkan, Sep 25 2015
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CROSSREFS
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Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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