Search: a244818 -id:a244818
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A244807
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The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.
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+10
12
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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
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OFFSET
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1,2
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COMMENTS
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Inspired by Stanislaw M. Ulam's hexagonal spiral, circa 1963. See example section of A056105.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A244811
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The hexagonal spiral of Champernowne, read along the 330-degree ray.
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+10
12
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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(3n^2 - 6n + 4)th almost natural number (A033307); also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A056107, A244807, A244808, A244809, A244810, A244812, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244812
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The hexagonal spiral of Champernowne, read along the 300-degree ray.
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+10
12
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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(12n^2 - 23n + 12)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A244804, A244807, A244808, A244809, A244810, A244811, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244808
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The hexagonal spiral of Champernowne, read along the 60-degree ray.
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+10
11
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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(12n^2-27n+16)th almost natural number (A033307); also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A244802, A244807, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244809
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The hexagonal spiral of Champernowne, read along the 30-degree ray.
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+10
11
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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(3n^2-7n+5)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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See A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A056106, A244807, A244808, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244810
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The hexagonal spiral of Champernowne, read along the North (or 360-degree) ray.
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+10
11
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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(12n^2 - 25n + 14)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A244803, A244807, A244808, A244809, A244811, A244812, A244813, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244813
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The hexagonal spiral of Champernowne, read along the West (or 270-degree) ray.
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+10
11
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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(3n^2 - 5n + 3)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A056108, A244807, A244808, A244809, A244810, A244811, A244812, A244814, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244814
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The hexagonal spiral of Champernowne, read along the 240-degree ray.
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+10
11
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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
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OFFSET
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1,3
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LINKS
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FORMULA
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(12n^2 - 21n + 10)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A244805, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244815, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244815
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The hexagonal spiral of Champernowne, read along the 210-degree ray.
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+10
11
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1, 6, 3, 2, 3, 4, 5, 8, 0, 1, 3, 6, 1, 4, 4, 2, 3, 3, 3, 0, 5, 4, 5, 8, 6, 8, 3, 7, 9, 9, 9, 1, 1, 1, 1, 1, 7, 1, 9, 1, 7, 1, 1, 1, 1, 1, 7, 1, 9, 2, 7, 2, 1, 2, 1, 2, 7, 2, 9, 2, 7, 3, 1, 3, 1, 3, 7, 3, 9, 3, 7, 4, 1, 4, 1, 4, 7, 4, 9, 4, 7, 5, 1, 5, 1, 5, 7, 5, 9, 6, 7, 6, 1, 6, 1, 7, 7, 7, 9, 7, 7, 7, 1, 8, 1
(list;
graph;
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listen;
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OFFSET
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1,2
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LINKS
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FORMULA
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(3n^2 - 4n + 2)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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 - 4n + 2 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]
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CROSSREFS
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Cf. A007376, A056109, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244816, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A244816
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The hexagonal spiral of Champernowne, read along the South (or 180-degree) ray.
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+10
11
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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
(list;
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listen;
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OFFSET
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1,3
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LINKS
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FORMULA
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(12n^2 - 19n + 8)th almost natural number (A033307), Also see formula section of A056105.
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EXAMPLE
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see A244807 example section for its diagram.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A007376, A244806, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244817, A244818.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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