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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

%I

%S 0,1,9,21,49,61,97,133,225,237,273,309,417,453,561,669,961,973,1009,

%T 1045,1153,1189,1297,1405,1729,1765,1873,1981,2305,2413,2737,3061,

%U 3969,3981,4017,4053,4161,4197,4305,4413,4737,4773,4881,4989,5313,5421,5745

%N Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

%C The structure has a fractal behavior similar to the toothpick sequence A139250.

%C First differences: A161415, where there is an explicit formula for the n-th term.

%C For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

%H 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 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca025.jpg">Illustration of initial terms</a>

%H 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 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 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = 1 + 4*A219954(n), n >= 1. - _M. F. Hasler_, Dec 02 2012

%F a(2^k) = (2^(k+1) - 1)^2. - _Omar E. Pol_, Jan 05 2013

%e From _Omar E. Pol_, Sep 24 2015: (Start)

%e With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:

%e 1;

%e 9;

%e 21, 49;

%e 61, 97, 133, 225;

%e 237, 273, 309, 417, 453, 561, 669, 961;

%e ...

%e Right border gives A060867.

%e 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 .

%e Illustration of initial terms, for n = 1..10:

%e . _ _ _ _ _ _ _ _

%e . | _ _ | | _ _ |

%e . | | _|_|_ _ _ _ _ _ _ _ _ _ _|_|_ | |

%e . | |_| _ _ _ _ _ _ _ _ |_| |

%e . |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|

%e . | |_| _ _ |_| |_| _ _ |_| |

%e . | | | _|_|_ _ _|_|_ | | |

%e . | _| |_| _ _ _ _ |_| |_ |

%e . | | |_ _| | _|_|_ | |_ _| | |

%e . | |_ _| | |_| _ |_| | |_ _| |

%e . | _ _ | _| |_| |_ | _ _ |

%e . | | _|_| | |_ _ _| | |_|_ | |

%e . | |_| _| |_ _| |_ _| |_ |_| |

%e . | | | |_ _ _ _ _ _ _| | | |

%e . | _| |_ _| |_ _| |_ _| |_ |

%e . _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _

%e . | _| |_ _| |_ _| |_ _| |_ _| |_ |

%e . | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |

%e . | |_ _| | | |_ _| |

%e . |_ _ _ _| |_ _ _ _|

%e .

%e After 10 generations there are 273 ON cells, so a(10) = 273.

%e (End)

%p read("transforms") ; isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:

%p A048883 := proc(n) 3^wt(n) ; end proc:

%p 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 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # _R. J. Mathar_, Oct 16 2010

%o (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 Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.

%K nonn,tabf

%O 0,3

%A _Omar E. Pol_, May 20 2009

%E Edited by _N. J. A. Sloane_, Jun 15 2009 and Jul 13 2009

%E More terms from _R. J. Mathar_, Oct 16 2010

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Last modified August 3 07:58 EDT 2021. Contains 346435 sequences. (Running on oeis4.)