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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).

%I A160414 #68 Sep 01 2023 08:17:48
%S A160414 0,1,9,21,49,61,97,133,225,237,273,309,417,453,561,669,961,973,1009,
%T A160414 1045,1153,1189,1297,1405,1729,1765,1873,1981,2305,2413,2737,3061,
%U A160414 3969,3981,4017,4053,4161,4197,4305,4413,4737,4773,4881,4989,5313,5421,5745
%N A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).
%C A160414 The structure has a fractal behavior similar to the toothpick sequence A139250.
%C A160414 First differences: A161415, where there is an explicit formula for the n-th term.
%C A160414 For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.
%H A160414 Paolo Xausa, <a href="/A160414/b160414.txt">Table of n, a(n) for n = 0..10000</a>
%H A160414 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A160414 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca025.jpg">Illustration of initial terms</a>
%H A160414 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca035.jpg">Illustration of the structure after 24th stage (contains 1729 ON cells)</a>
%H A160414 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A160414 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A160414 a(n) = 1 + 4*A219954(n), n >= 1. - _M. F. Hasler_, Dec 02 2012
%F A160414 a(2^k) = (2^(k+1) - 1)^2. - _Omar E. Pol_, Jan 05 2013
%e A160414 From _Omar E. Pol_, Sep 24 2015: (Start)
%e A160414 With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
%e A160414 1;
%e A160414 9;
%e A160414 21,    49;
%e A160414 61,    97,  133,  225;
%e A160414 237,  273,  309,  417,  453, 561,  669,  961;
%e A160414 ...
%e A160414 Right border gives A060867.
%e A160414 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.
%e A160414 .
%e A160414 Illustration of initial terms, for n = 1..10:
%e A160414 .       _ _ _ _                       _ _ _ _
%e A160414 .      |  _ _  |                     |  _ _  |
%e A160414 .      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
%e A160414 .      | |_|  _ _     _ _   _ _     _ _  |_| |
%e A160414 .      |_ _| |  _|_ _|_  | |  _|_ _|_  | |_ _|
%e A160414 .          | |_|  _ _  |_| |_|  _ _  |_| |
%e A160414 .          |   | |  _|_|_ _ _|_|_  | |   |
%e A160414 .          |  _| |_|  _ _   _ _  |_| |_  |
%e A160414 .          | | |_ _| |  _|_|_  | |_ _| | |
%e A160414 .          | |_ _| | |_|  _  |_| | |_ _| |
%e A160414 .          |  _ _  |  _| |_| |_  |  _ _  |
%e A160414 .          | |  _|_| | |_ _ _| | |_|_  | |
%e A160414 .          | |_|  _| |_ _| |_ _| |_  |_| |
%e A160414 .          |   | | |_ _ _ _ _ _ _| | |   |
%e A160414 .          |  _| |_ _| |_   _| |_ _| |_  |
%e A160414 .       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
%e A160414 .      |  _| |_ _|   |_ _| |_ _|   |_ _| |_  |
%e A160414 .      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A160414 .      | |_ _| |                     | |_ _| |
%e A160414 .      |_ _ _ _|                     |_ _ _ _|
%e A160414 .
%e A160414 After 10 generations there are 273 ON cells, so a(10) = 273.
%e A160414 (End)
%p A160414 read("transforms") ; isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
%p A160414 A048883 := proc(n) 3^wt(n) ; end proc:
%p A160414 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:
%p A160414 A160414 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # _R. J. Mathar_, Oct 16 2010
%t A160414 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_ *)
%o A160414 (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
%Y A160414 Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.
%K A160414 nonn,tabf
%O A160414 0,3
%A A160414 _Omar E. Pol_, May 20 2009
%E A160414 Edited by _N. J. A. Sloane_, Jun 15 2009 and Jul 13 2009
%E A160414 More terms from _R. J. Mathar_, Oct 16 2010