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 A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4). 18
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.] Omar E. Pol, Illustration of initial terms N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS FORMULA 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 EXAMPLE From Omar E. Pol, Sep 24 2015: (Start) 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; ... Right border gives A060867. 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. . Illustration of initial terms, for n = 1..10: .       _ _ _ _                       _ _ _ _ .      |  _ _  |                     |  _ _  | .      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | | .      | |_|  _ _     _ _   _ _     _ _  |_| | .      |_ _| |  _|_ _|_  | |  _|_ _|_  | |_ _| .          | |_|  _ _  |_| |_|  _ _  |_| | .          |   | |  _|_|_ _ _|_|_  | |   | .          |  _| |_|  _ _   _ _  |_| |_  | .          | | |_ _| |  _|_|_  | |_ _| | | .          | |_ _| | |_|  _  |_| | |_ _| | .          |  _ _  |  _| |_| |_  |  _ _  | .          | |  _|_| | |_ _ _| | |_|_  | | .          | |_|  _| |_ _| |_ _| |_  |_| | .          |   | | |_ _ _ _ _ _ _| | |   | .          |  _| |_ _| |_   _| |_ _| |_  | .       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _ .      |  _| |_ _|   |_ _| |_ _|   |_ _| |_  | .      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | .      | |_ _| |                     | |_ _| | .      |_ _ _ _|                     |_ _ _ _| . After 10 generations there are 273 ON cells, so a(10) = 273. (End) MAPLE 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) ; # R. J. Mathar, Oct 16 2010 PROG (PARI) my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<

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Last modified October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)