A033307 -id:A033307 - cp's OEIS frontend

A033989 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 negative x-axis.

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

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 leftwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033990. Spiral without zero: A033952.

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

a033989 0 = 0
a033989 n = a033307 $ a185950 n  -- Reinhard Zumkeller, Aug 14 2013

nmax = 1000; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2+10*nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2-n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 23 2017, after Andrew Woods *)

a(n) = A033307(4*n^2-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

A031312 Successive digits of odd numbers.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

The constant 0.135791113... whose decimal expansion is this sequence (analogous to Champernowne constant, A033307) is a transcendental number but is not a Liouville number (Wananiyakul et al., 2022). - Amiram Eldar, Feb 05 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Saeree Wananiyakul, Vichian Laohakosol and Janyarak Tongsomporn, Transcendental Numbers Arising from Mahler's Method, Mathematical, Computational Intelligence and Engineering Approaches for Tourism, Agriculture and Healthcare, Springer, Singapore, 2022, pp. 111-120.

Cf. A005408, A033307.

Flatten[IntegerDigits[2*Range[60] - 1]] (* Stefan Steinerberger, Apr 14 2006 *)

More terms from Stefan Steinerberger, Apr 14 2006

A244688 The spiral of Champernowne read by the South-Southwest ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

			See A244677 for the spiral of David Gawen Champernowne.

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

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]

See A244677 formula section.

A244690 The spiral of Champernowne read by the South-Southeast ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

			See A244677 for the spiral of David Gawen Champernowne.

Cf. A033307, A143839, A033952, A244677-A244688, A244691, A244692.

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 - 25n + 10 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

See A244677 formula section.

A244692 The spiral of Champernowne read by the East-Southeast ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

			See A244677 for the spiral of David Gawen Champernowne.

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

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]

See A244677 formula section.

A031324 Decimal digits of successive Fibonacci numbers.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Decimal concatenation of Fibonacci numbers in base 10. - Daniel Forgues, Mar 25 2018

			0.011235813213455891442333776109871597...

Robert Israel, Table of n, a(n) for n = 0..10000
Brennan Benfield and Michelle Manes, The Fibonacci Sequence is Normal Base 10, arXiv:2202.08986 [math.NT], 2022.

Cf. A000045, A021093, A031325 - A031334.

Cf. A033307, A033308.

F:= [seq(combinat:-fibonacci(n),n=0..50)]:
map(t -> op(ListTools:-Reverse(convert(t,base,10))),F); # Robert Israel, Oct 11 2024

Flatten[IntegerDigits/@Fibonacci[Range[0,30]]] (* Harvey P. Dale, Jan 28 2015 *)

An approximation, where each successive Fibonacci number is shifted right by one place (thus causing an overlap when numbers have more than one digit), is given by 10/89 (A021093). - Daniel Forgues, Mar 25 2018

A048992 Hannah Rollman's numbers: the numbers excluded from A048991.

Original entry on oeis.org

12, 23, 31, 34, 41, 42, 45, 51, 52, 53, 56, 61, 62, 63, 64, 67, 71, 72, 73, 74, 75, 78, 81, 82, 83, 84, 85, 86, 89, 91, 92, 93, 94, 95, 96, 97, 98, 101, 111, 113, 121, 122, 123, 131, 141, 151, 161, 171, 181, 191, 192, 201, 202, 210, 211, 212, 213, 214, 215, 216, 217

Offset: 1

Bernardo Recamán

A105390(n) = number of terms <= n; for n < 740: A105390(n) < n/2. - Reinhard Zumkeller, Apr 04 2005
A116700 is a similar sequence. Note that 21 is missing from the current sequence, because we deleted 12 in computing A048991 and now 21 is no longer "earlier in the sequence". On the other hand 21 is present in A116700. - N. J. A. Sloane, Aug 05 2007
Otherwise said: Numbers which occur in the concatenation of all smaller numbers not listed in this sequence. - M. F. Hasler, Dec 29 2012
Number of terms < 10^n, n = 1, 2, ...: (0, 37, 589, 7046, ...), gives number of n-digit terms as first differences: (37, 552, 6457, ...). - M. F. Hasler, Oct 25 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Nick Hobson, Python program for this sequence

Complement of A048991.

Similar to A116700: "early birds" in the Barbier word A007376 or Champernowne sequence A033307.

import Data.List (isInfixOf)
a048992 n = a048992_list !! (n-1)
a048992_list = g [1..] [] where
   g (x:xs) ys | xs' `isInfixOf` ys = x : g xs ys
               | otherwise          = g xs (xs' ++ ys)
               where xs' = reverse $ show x
-- Reinhard Zumkeller, Dec 05 2011

a[0] = 1; s = "1"; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, If[StringFreeQ[s, t = ToString[k]], s = s <> t, Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 25 2013 *)

D=[]; for(n=1, 999, for(i=0, #D-#d=digits(n), D[i+1..i+#d]!=d || print1(n",") || next(2)); D=concat(D, d)) \\ M. F. Hasler, Oct 25 2019

Python
```
# see Hobson link
```

Edited by Patrick De Geest, Jun 02 2003

A033990 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 negative y-axis.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Consider array of digits 0_(1)23456789(1)0111213141516171(8)1920212223...; in this array add to n-th pointer 8*n+1 to get next pointer. E.g., n=1 so n+(8*1+1)=10 -> n=10 so n+(8*2+1)=27 -> n=27 so ... etc. - comment from Patrick De Geest.

			The spiral begins
                 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
                                         |
                             3---0---3---9
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading downwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033989. 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 Patrick De Geest, Oct 15 1999

Edited by Charles R Greathouse IV, Nov 01 2009

A077771 Decimal value of the ternary Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 15 2002

The first 99 digits form a prime. - Jonathan Vos Post, Nov 11 2004

This constant is 3-normal. - Charles R Greathouse IV, Feb 06 2015

			0.598958167538433992500172217929...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary Champernowne Constant

Cf. A054635 (base 3 digits), A077772 (continued fraction).

Cf. A030190, A066716, A066717: binary digits, decimals and continued fraction of the binary Champernowne constant; A033307: decimal Champernowne constant.

First[RealDigits[ChampernowneNumber[3], 10, 100]] (* Paolo Xausa, May 03 2024 *)

A077771(b=3,t=1.,s=b)={sum(n=1, default(realprecision)*2.303\log(b)+1, nM. F. Hasler, Oct 25 2019

A131881 Complement of A116700. Might be called "punctual birds".

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55, 57, 58, 59, 60, 66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107, 108, 109, 113, 114

Offset: 1

M. F. Hasler, Jul 23 2007

Numbers n that do not occur in the concatenation of 1,2,3...,n-1.
Every power of 10 is a member, which proves that the sequence is infinite. - N. J. A. Sloane, Jul 23 2007
The asymptotic density of the sequence is zero. The number of k-digit terms is A132133 = (9, 45, 270, 2104, ...), k = 1, 2, .... These are the first difference of the indices of powers of 10, T = (1, 10, 55, 325, 2429, ...), which we get as partial sums if we prefix A132133(0) = 1 corresponding to the number 0. - M. F. Hasler, Oct 24 2019

			The first number not in this sequence is the early bird "12" which occurs as concatenation of 1 and 2.

M. F. Hasler, Table of n, a(n) for n = 1..2428

Cf. A116700 (early birds), A132133 (number of n-digit terms).

Cf. A007376 (Barbier word ...,8,9,1,0,1,1,...), A033307 (Champernowne constant).

$s="0"; for(; ++$i < 2000; $s .= $i) if( !strpos($s,"$i")) echo $i,", ";

cp's OEIS Frontend

A033989 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 negative x-axis.

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A031312 Successive digits of odd numbers.

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Mathematica

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A244688 The spiral of Champernowne read by the South-Southwest ray.

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Mathematica

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A244690 The spiral of Champernowne read by the South-Southeast ray.

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Mathematica

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A244692 The spiral of Champernowne read by the East-Southeast ray.

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Mathematica

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A031324 Decimal digits of successive Fibonacci numbers.

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Maple

Mathematica

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A048992 Hannah Rollman's numbers: the numbers excluded from A048991.

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A033990 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 negative y-axis.

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Author

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Comments