A244815 -id:A244815 - cp's OEIS frontend

A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

Inspired by Stanislaw M. Ulam's hexagonal spiral, circa 1963. See example section of A056105.

When A056105, A056106, A056107, A056108, A056109 & A003215 were submitted, the offsets were 0. Here the offset is 1.

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..................7...5...1...6...5...1...5...5...1...4
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................1...6...3...1...5...3...1...4...3...1...3
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..............3...1...7...1...1...6...1...1...5...1...1...3
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............7...1...1...0...0...1...9...9...8...9...7...4...1
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..........1...8...0...7...8...7...7...7...6...7...5...9...1...2
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........3...1...1...9...9...5...8...5...7...5...6...7...6...1...3
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......8...1...1...8...6...4...2...4...1...4...0...5...4...9...3...1
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....1...9...0...0...0...3...9...2...8...2...7...4...5...7...5...1...1
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..3...1...2...8...6...4...3...1...8...1...7...2...9...5...3...9...1...3
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9...2...1...1...1...4...0...9...1...1...0...1...6...3...4...7...4...2...1
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..0...0...8...6...4...3...2...1...4...3...1...6...2...8...5...2...9...1...0
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1...3...2...2...5...1...0...2...5...1...2...9...1...5...3...3...7...3...1...3
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..2...1...8...6...4...3...2...1...6...7...8...5...2...7...5...1...9...1...1
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....1...0...3...3...6...2...1...3...1...4...1...4...3...2...7...2...1...9
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......1...4...8...6...4...3...2...2...2...3...2...6...5...0...9...1...2
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........2...1...4...4...7...3...3...4...3...5...3...1...7...1...0...1
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..........2...0...8...6...4...8...4...9...5...0...5...9...9...1...8
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............1...5...5...5...6...6...6...7...6...8...6...0...1...2
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..............2...1...8...6...8...7...8...8...8...9...9...9...1
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................3...0...6...1...0...7...1...0...8...1...0...7
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..................1...2...4...1...2...5...1...2...6...1...2
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....................1...4...4...1...4...5...1...4...6...1
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Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 3n^2- 8n +6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

For each 30 degrees of the compass, the corresponding spoke (or ray) has a generating formula as follows:
 3n^2- 8n +6
12n^2-27n+16
 3n^2- 7n+ 5
12n^2-25n+14
 3n^2 -6n +4
12n^2-23n+12
 3n^2 -5n +3
12n^2-21n+10
 3n^2 -4n +2
12n^2-19n +8
 3n^2 -3n +1
12n^2-17n+ 6
Also see formula section of A056105.

A244811 The hexagonal spiral of Champernowne, read along the 330-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A056107, A244807, A244808, A244809, A244810, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 3n^2 - 6n + 4 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 6n + 4)th almost natural number (A033307); also see formula section of A056105.

A244812 The hexagonal spiral of Champernowne, read along the 300-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244804, A244807, A244808, A244809, A244810, A244811, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 12n^2 - 23n + 12 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 23n + 12)th almost natural number (A033307), Also see formula section of A056105.

A244808 The hexagonal spiral of Champernowne, read along the 60-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244802, A244807, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 12n^2 - 27n + 16 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2-27n+16)th almost natural number (A033307); also see formula section of A056105.

A244809 The hexagonal spiral of Champernowne, read along the 30-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A244807 example section for its diagram.

Cf. A007376, A056106, A244807, A244808, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 3n^2- 7n+ 5 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2-7n+5)th almost natural number (A033307), Also see formula section of A056105.

A244810 The hexagonal spiral of Champernowne, read along the North (or 360-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244803, A244807, A244808, A244809, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

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_] := 12n^2 - 25n + 14 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 25n + 14)th almost natural number (A033307), Also see formula section of A056105.

A244813 The hexagonal spiral of Champernowne, read along the West (or 270-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A056108, A244807, A244808, A244809, A244810, A244811, A244812, A244814, A244815, A244816, A244817, A244818.

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_] := 3n^2 - 5n + 3 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 5n + 3)th almost natural number (A033307), Also see formula section of A056105.

A244814 The hexagonal spiral of Champernowne, read along the 240-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244805, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244815, A244816, A244817, A244818.

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_] := 12n^2 - 21n + 10 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 21n + 10)th almost natural number (A033307), Also see formula section of A056105.

A244816 The hexagonal spiral of Champernowne, read along the South (or 180-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244806, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244817, A244818.

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_] := 12n^2 - 19n + 8 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 19n + 8)th almost natural number (A033307), Also see formula section of A056105.

A244817 The hexagonal spiral of Champernowne, read along the 150-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A003215, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244818.

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_] := 3n^2 - 3n + 1 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 3n + 1)th almost natural number (A033307), Also see formula section of A056105.

cp's OEIS Frontend

A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

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A244811 The hexagonal spiral of Champernowne, read along the 330-degree ray.

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A244812 The hexagonal spiral of Champernowne, read along the 300-degree ray.

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A244808 The hexagonal spiral of Champernowne, read along the 60-degree ray.

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A244809 The hexagonal spiral of Champernowne, read along the 30-degree ray.

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A244810 The hexagonal spiral of Champernowne, read along the North (or 360-degree) ray.

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A244813 The hexagonal spiral of Champernowne, read along the West (or 270-degree) ray.

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A244814 The hexagonal spiral of Champernowne, read along the 240-degree ray.

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A244816 The hexagonal spiral of Champernowne, read along the South (or 180-degree) ray.

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