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A244677 The spiral of Champernowne, read along the East ray. 24
1, 2, 0, 1, 1, 4, 8, 9, 1, 1, 6, 8, 2, 4, 8, 3, 6, 0, 4, 9, 5, 6, 6, 1, 7, 4, 1, 9, 0, 1, 1, 1, 7, 1, 4, 7, 6, 1, 6, 6, 7, 1, 0, 9, 0, 2, 3, 5, 5, 2, 7, 4, 2, 3, 1, 6, 1, 3, 5, 1, 2, 3, 0, 9, 5, 4, 5, 1, 0, 4, 1, 6, 7, 5, 6, 4, 6, 6, 3, 5, 7, 6, 9, 0, 0, 7, 6, 8, 5, 8, 3, 9, 2, 8, 0, 3, 1, 9, 8, 0, 0, 3, 0, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inspired by Stanislaw Ulam's spiral, circa 1963.

LINKS

Table of n, a(n) for n=1..105.

Robert G. Wilson v, Cover of the March 1964 issue of Scientific American

FORMULA

Formulas for rays in directions of 32 compass points:

  SE     4n^2   -4n  +1

  SExS  64n^2 -113n +50

  SSE   16n^2  -25n +10

  SxE   64n^2 -115n +52

  S      4n^2   -5n  +2

  SxW   64n^2 -117n +54

  SSW   16n^2  -27n +12

  SWxS  64n^2 -119n +56

  SW     4n^2   -6n  +3

  SWxW  64n^2 -121n +58

  WSW   16n^2  -29n +14

  WxS   64n^2 -123n +60

  W      4n^2   -7n  +4

  WxN   64n^2 -125n +62

  WNW   16n^2  -31n +16

  NWxW  64n^2 -127n +64

  NW     4n^2   -8n  +5

  NWxN  64n^2 -129n +66

  NNW   16n^2  -33n +18

  NxW   64n^2 -131n +68

  N      4n^2   -9n  +6

  NxE   64n^2 -133n +70

  NNE   16n^2  -35n +20

  NExN  64n^2 -135n +72

  NE     4n^2  -10n  +7

  NExE  64n^2 -137n +74

  ENE   16n^2  -37n +22

  ExN   64n^2 -139n +76

  E      4n^2  -11n  +8

  ExS   64n^2 -141n +78

  ESE   16n^2  -39n +24

  SExE  64n^2 -143n +80

EXAMPLE

The beginning of the infinite spiral of David Gawen Champernowne:

.

  7--1--9--6--1--8--6--1--7--6--1--6--6--1--5--6--1--4--6--1--3  .

  |                                                           |  |

  0  1--4--4--1--3--4--1--2--4--1--1--4--1--0--4--1--9--3--1  6  .

  |  |                                                     |  |  |

  1  4  2--1--1--2--1--0--2--1--9--1--1--8--1--1--7--1--1  8  1  .

  |  |  |                                               |  |  |  |

  7  5  2  0--1--1--0--1--0--0--1--9--9--8--9--7--9--6  6  3  2  9

  |  |  |  |                                         |  |  |  |  |

  1  1  1  2  7--7--6--7--5--7--4--7--3--7--2--7--1  9  1  1  6  8

  |  |  |  |  |                                   |  |  |  |  |  |

  1  4  2  1  7  5--5--4--5--3--5--2--5--1--5--0  7  5  1  7  1  1

  |  |  |  |  |  |                             |  |  |  |  |  |  |

  7  6  3  0  8  5  7--3--6--3--5--3--4--3--3  5  0  9  5  3  1  8

  |  |  |  |  |  |  |                       |  |  |  |  |  |  |  |

  2  1  1  3  7  6  3  3--2--2--2--1--2--0  3  9  7  4  1  1  6  8

  |  |  |  |  |  |  |  |                 |  |  |  |  |  |  |  |  |

  1  4  2  1  9  5  8  2  3--1--2--1--1  2  2  4  9  9  1  6  1  1

  |  |  |  |  |  |  |  |  |           |  |  |  |  |  |  |  |  |  |

  7  7  4  0  8  7  3  4  1  5--4--3  1  9  3  8  6  3  4  3  0  7

  |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |  |  |

  3  1  1  4  0  5  9  2  4  6  1--2  0  1  1  4  8  9  1  1  6  8

  |  |  |  |  |  |  |  |  |  |        |  |  |  |  |  |  |  |  |  |

  1  4  2  1  8  8  4  5  1  7--8--9--1  8  3  7  6  2  1  5  1  1

  |  |  |  |  |  |  |  |  |              |  |  |  |  |  |  |  |  |

  7  8  5  0  1  5  0  2  5--1--6--1--7--1  0  4  7  9  3  3  9  6

  |  |  |  |  |  |  |  |                    |  |  |  |  |  |  |  |

  4  1  1  5  8  9  4  6--2--7--2--8--2--9--3  6  6  1  1  1  5  8

  |  |  |  |  |  |  |                          |  |  |  |  |  |  |

  1  4  2  1  2  6  1--4--2--4--3--4--4--4--5--4  6  9  1  4  1  1

  |  |  |  |  |  |                                |  |  |  |  |  |

  7  9  6  0  8  0--6--1--6--2--6--3--6--4--6--5--6  0  2  3  8  5

  |  |  |  |  |                                      |  |  |  |  |

  5  1  1  6  3--8--4--8--5--8--6--8--7--8--8--8--9--9  1  1  5  8

  |  |  |  |                                            |  |  |  |

  1  5  2  1--0--7--1--0--8--1--0--9--1--1--0--1--1--1--1  3  1  1

  |  |  |                                                  |  |  |

  7  0  7--1--2--8--1--2--9--1--3--0--1--3--1--1--3--2--1--3  7  4

  |  |                                                        |  |

  6  1--5--1--1--5--2--1--5--3--1--5--4--1--5--5--1--5--6--1--5  8

  |                                                              |

  1--7--7--1--7--8--1--7--9--1--8--0--1--8--1--1--8--2--1--8--3--1

MATHEMATICA

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 4n^2 - 11n + 8 (* see formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A054552, A244678 - A244688, A033952, A244690 - A244692.

Sequence in context: A306800 A235955 A077762 * A243986 A322838 A085496

Adjacent sequences:  A244674 A244675 A244676 * A244678 A244679 A244680

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

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Last modified January 23 15:15 EST 2021. Contains 340385 sequences. (Running on oeis4.)