A033307 -id:A033307 - cp's OEIS frontend

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

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.

A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			  2---3---2---4---2---5---2
  |                       |
  2   1---3---1---4---1   6
  |   |               |   |
  2   2   4---5---6   5   2
  |   |   |       |   |   |
  1   1   3   0   7   1   7
  |   |   |   |   |   |   |
  2   1   2---1   8   6   2
  |   |           |   |   |
  0   1---0---1---9   1   8
  |                   |   |
  2---9---1---8---1---7   2
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading rightwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Sequences based on the same spiral: A033988, A033989, A033990. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]];
a[n_] := If[n==0, 0, A033307[[4n^2 + 3n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + 3*n - 1) for n > 0. - Andrew Woods, May 20 2012

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A066547 Let N = 123456789101112131415161718..., the concatenation of the natural numbers. a(n) is the n-digit number formed from the digits of N starting from the {n(n-1)/2 +1}th digit. Omit any leading zeros.

Original entry on oeis.org

1, 23, 456, 7891, 1112, 131415, 1617181, 92021222, 324252627, 2829303132, 33343536373, 839404142434, 4454647484950, 51525354555657, 585960616263646, 5666768697071727, 37475767778798081, 828384858687888990, 9192939495969798991, 101102103104105106, 107108109110111112113

Offset: 1

Amarnath Murthy, Dec 16 2001

			1, 23, 456, 7891, 01112, 131415, 1617181, 92021222, 3... becomes 1, 23, 456, 7891, 1112, 131415, 1617181, 92021222, ...

Jason Bard, Table of n, a(n) for n = 1..1000 (first 71 terms from Vincenzo Librandi)

Cf. A100751, A007376, A033307, A066548.

Cf. A001369, A007923.

d = Flatten[IntegerDigits /@ Range[90]]; Table[FromDigits[Take[d, {n(n + 1)/2 + 1, (n + 1)(n + 2)/2}]], {n, 0, 17}] (* Robert G. Wilson v, Nov 22 2004 *)

N=[]; k=0; for(n=1,20, while(#NM. F. Hasler, May 08 2014

More terms from Larry Reeves (larryr(AT)acm.org), Dec 18 2001

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.

A244815 The hexagonal spiral of Champernowne, read along the 210-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A056109, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, 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 - 4n + 2 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

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

cp's OEIS Frontend

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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A033953 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

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A066547 Let N = 123456789101112131415161718..., the concatenation of the natural numbers. a(n) is the n-digit number formed from the digits of N starting from the {n(n-1)/2 +1}th digit. Omit any leading zeros.

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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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