A244690 -id:A244690

A244677

The spiral of Champernowne, read along the East ray.

+10
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.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244688

The spiral of Champernowne read by the South-Southwest ray.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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_] := 16n^2 - 27n + 12 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A143838, A033952, A244677-A244687, A244690-A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244692

The spiral of Champernowne read by the East-Southeast ray.

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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_] := 16n^2 - 39n + 24 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A143855, A033952, A244677-A244688, A244690, A244691.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244691

The spiral of Champernowne read by the Southeast ray.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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 - 4n + 1 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A016754, A033952, A244677-A244688, A244690, A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244678

The spiral of Champernowne read by the East-Northeast ray.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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_] := 16n^2 - 37n + 22 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A143856, A244677 - A244688, A033952, A244690 - A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244687

The spiral of Champernowne read by the Southwest ray.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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 - 6n + 3 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A054569, A033952, A244677-A244688, A244690-A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244679

The spiral of Champernowne read by the Northeast ray.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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 - 10n + 7 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A054554, A244677 - A244688, A033952, A244690 - A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244680

The spiral of Champernowne read by the North-Northeast ray.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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_] := 16n^2 - 35n + 20 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A033952, A143861, A244677 - A244688, A244690 - A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244681

The spiral of Champernowne read by the North ray.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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 - 9n + 6 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A054556, A033952, A244677 - A244688, A244690 - A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved

A244682

The spiral of Champernowne read by the North-Northwest ray.

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

See A244677 formula section.

EXAMPLE

See A244677 for the spiral of David Gawen Champernowne.

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_] := 16n^2 - 33n + 18 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

CROSSREFS

Cf. A033307, A033952, A143860, A244677-A244688, A244690-A244692.

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v, Jul 04 2014

STATUS

approved