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A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition). 6
0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.

At stage 1, only one cell is turned ON, so a(1) = 1.

If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.

The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).

Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).

Has the same rules as A256534 but here a(1) = 1 not 4.

Has a smoother behavior than A160414 with which shares infinitely many terms (see example).

A256531, the first differences, gives the number of cells turned ON at n-th stage.

LINKS

Table of n, a(n) for n=0..57.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

FORMULA

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

EXAMPLE

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,  157,  225;

237,  273,  333,  417,  525,  657,  813,  961;

...

Right border gives A060867.

This triangle T(n,k) shares with the triangle A160414 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.

PROG

(GWBASIC) 10' a256530 First 2^z-1 terms: 20 z=6: defdbl a: for i=1 to z: for j=0 to 2^(i-1)-1: n=n+1: a(n)=(2^i-1)^2 + 3*(2*j)^2: print a(n); : next j: next i: end

CROSSREFS

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

Sequence in context: A241747 A133762 A160414 * A118130 A144482 A251212

Adjacent sequences:  A256527 A256528 A256529 * A256531 A256532 A256533

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Apr 21 2015

STATUS

approved

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Last modified June 19 15:03 EDT 2021. Contains 345141 sequences. (Running on oeis4.)