A160414 - OEIS

A160414

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

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

Paolo Xausa, Table of n, a(n) for n = 0..10000

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

Omar E. Pol, Illustration of the structure after 24th stage (contains 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

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:

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A259243 A241747 A133762 * A256530 A363634 A118130

Adjacent sequences: A160411 A160412 A160413 * A160415 A160416 A160417

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, May 20 2009

EXTENSIONS

Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009

More terms from R. J. Mathar, Oct 16 2010

STATUS

approved