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

Author

Robert G. Wilson v, Jul 06 2014

Keywords

• nonn
• base
• easy

Comments

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.

Examples

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

Crossrefs

Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.

Programs

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

Formula

For each 30 degrees of the compass, the corresponding spoke (or ray) has a generating formula as follows:

090: 3n^2- 8n +6

060: 12n^2-27n+16

030: 3n^2- 7n+ 5

000: 12n^2-25n+14

330: 3n^2 -6n +4

300: 12n^2-23n+12

270: 3n^2 -5n +3

240: 12n^2-21n+10

210: 3n^2 -4n +2

180: 12n^2-19n +8

150: 3n^2 -3n +1

120: 12n^2-17n+ 6

Also see formula section of A056105.