cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A101403 Number of times that n occurs in A101402.

Original entry on oeis.org

1, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 4, 4, 3, 2, 4, 3, 2, 3, 3
Offset: 0

Views

Author

Odimar Fabeny, Jan 16 2005

Keywords

Comments

First differences of A246439; see A246431 for smallest m such that a(m)=n; a(A129117(n)) = 5. - Reinhard Zumkeller, Aug 28 2014

Links

Crossrefs

Cf. A101402, A246431, A246439.

Programs

  • Haskell
    import Data.List (group)
a101403 n = a101403_list !! n
a101403_list = map length $ group a101402_list
-- Reinhard Zumkeller, Aug 27 2014
  • Mathematica
    A101402[0] = 0; A101402[1] = 1;
A101402[n_] := A101402[n] = A101402[2^(Floor[Log[2, n - 1]])] + A101402[n - 1 - 2^(Floor[Log[2, n - 1]])]; TheList = Table[A101402[i], {i, 0, 279}];
A101403 = Table[Count[TheList, i], {i, 0, Last[TheList]}]
(* Keith Schneider, May 25 2007 *)

Formula

a(n) > 1 for all n > 0, see comment in A101402. - Charles R Greathouse IV, Sep 10 2014

Extensions

More terms from Keith Schneider, May 25 2007

A129117 Numbers that appear exactly five times in A101402. (Also indices of fives in A101403.).

Original entry on oeis.org

727, 2181, 3635, 5089, 6543, 7997, 9451, 10905, 12358, 13812, 15266, 16720, 18174, 19628, 21082, 22536, 23989, 25443, 26897, 28351, 29805, 31259, 32713, 34167, 35620, 37074, 38528, 39982, 41436, 42890, 44344, 45798, 47250, 48704, 50158
Offset: 1

Views

Author

Keith Schneider (schneidk(AT)email.unc.edu), May 25 2007

Keywords

Comments

It is also interestng to look at this sequence modulo 727.

Examples

			a(1) = 727 since A101402(2045) = A101402(2046) = A101402(2047) = A101402(2048) = A101402(2049) = 727.
a(1) = 727 since A101403(727) = 5.

Links

Crossrefs

Cf. A101402, A101403.

Programs

  • Haskell
    import Data.List (elemIndices)
a129117 n = a129117_list !! (n-1)
a129117_list = elemIndices 5 a101403M_list
-- Reinhard Zumkeller, Aug 28 2014
  • Mathematica
    A101402[0] = 0; A101402[1] = 1; A101402[n_] := A101402[n] = A101402[2^(Floor[Log[2, n - 1]])] + A101402[n - 1 - 2^(Floor[Log[2, n - 1]])]; TheList = Table[A101402[i], {i, 0, 203000}]; TheList2 = Union[TheList]; A101403 = Table[Count[TheList, i], {i, 0, Last[TheList]}]; TheSeq = Delete[Union[Table[ If[TheList[[i]] == TheList[[i + 4]], TheList[[i]]], {i, 1, Length[TheList] - 4}]], -1] Count[A101403, 5] Length[TheSeq]

Formula

A101403(a(n)) = 5.

A247562 A101402(10^n).

Original entry on oeis.org

1, 4, 36, 355, 3549, 35495, 354942, 3549413, 35494123, 354941215, 3549412151, 35494121507, 354941215066, 3549412150655, 35494121506548, 354941215065477, 3549412150654758, 35494121506547571, 354941215065475694, 3549412150654756948
Offset: 0

Views

Author

Charles R Greathouse IV and Robert G. Wilson v, Sep 20 2014

Keywords

Comments

if c = 0.3549412150654756947265447261312365125932387..., then a(n) =~ floor(c*10^n). This holds to n^7 +- 2.

Links

Crossrefs

Cf. A101402.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := a[n] = Block[{p = 2^(Ceiling[Log[2, n]] - 1)}, a[p] + a[n - 1 - p]];
Table[a[10^n],{n,0,20}]

A101266 First differences of A101402.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0
Offset: 0

Views

Author

Odimar Fabeny, Jan 23 2005

Keywords

Comments

All terms are 0 or 1, see comment in A101402. - Charles R Greathouse IV, Aug 27 2014

Crossrefs

Cf. A101402.

Programs

  • Maple
    A101402 := proc(n) option remember; local k; if n <=1 then n; else k := 2^ceil(log[2](n)) ; procname(k/2)+procname(n-1-k/2) ; fi; end: A101266 := proc(n) A101402(n+1)-A101402(n) ; end: seq(A101266(n),n=0..130) ; # R. J. Mathar, Aug 17 2009
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := a[n] = Block[{p = 2^(Ceiling[ Log[2, n]] - 1)}, a[p] + a[n - 1 - p]]; t = Table[ a@n, {n, 0, 101}]; Rest@t - Most@t (* Robert G. Wilson v, Aug 17 2009 *)

Extensions

More terms from R. J. Mathar, Aug 17 2009

A101433 Partial sums of A101402.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 13, 16, 19, 23, 27, 31, 36, 41, 47, 53, 59, 66, 73, 80, 88, 96, 105, 114, 123, 132, 142, 152, 162, 173, 184, 196, 208, 221, 234, 247, 261, 275, 290, 305, 320, 335, 351, 367, 383, 400, 417, 435, 453, 471, 490, 509, 528, 548, 568, 589, 610, 631, 652, 674, 696, 718, 741, 764, 787, 811, 835
Offset: 0

Views

Author

Odimar Fabeny, Jan 17 2005

Keywords

Links

Crossrefs

Cf. A101402.

Extensions

Definition and more terms from Charles R Greathouse IV, Sep 10 2014

A164349 The limit of the string "0, 1" under the operation 'repeat string twice and remove last symbol'.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1
Offset: 0

Views

Author

Jack W Grahl, Aug 13 2009

Keywords

Comments

We start with the string 01, at each step we replace the string by two concatenated copies and remove the last symbol.
01 -> 010 -> 01001 -> 010010100 etc.
Each string consists of 2^n + 1 symbols and clearly after this step the first 2^n + 1 symbols do not change.
Equivalently this sequence is given as follows: a(0) = 0, a(1) = 1, and for n>1, a(n) = first one of g(n), g(g(n)), g(g(g(n))).. etc. to be either 0 or 1, where g(n) = A053645(n-1).
The proportion of 0's in this sequence converges to a number close to 0.645059. The exact nature of this constant is not known. It is easy to show there are no 2 consecutive 1's.
Start of the first occurrence of k consecutive zeros: 0, 2, 7, 2046, > 8388600, ..., . - Robert G. Wilson v, Aug 17 2009
Start of the first occurrence of 5 consecutive zeros is > 2^34 - 5. Sum of the first 10^n terms b(n) begins: 0, 3, 36, 355, 3549, 35494, 354942, 3549412, 35494122, 354941215, 3549412151. - Alex Ratushnyak, Aug 15 2012
a(A246439(n)) = 1; a(A246438(n)) = 0. - Reinhard Zumkeller, Aug 28 2014
The partial sums appear to give A101402. - Arie Groeneveld, Aug 27 2014

Links

Crossrefs

Cf. A053645, A246438, A246439, A101402.

Programs

  • Haskell
    a164349 n = if n == 0 then 0 else until (<= 1) (a053645 . subtract 1) n
-- Reinhard Zumkeller, Aug 28 2014
  • Mathematica
    Nest[ Most@ Flatten@ {#, #} &, {0, 1}, 7] (* Robert G. Wilson v, Aug 17 2009 *)
  • Perl
    my $ab = "10"; for (my $j = 1; $j < 30; $j++) { $ab .= $ab; substr $ab, -1, 1, ""; print "$ab\n"; }

Extensions

Spelling and notation corrected by Charles R Greathouse IV, Mar 23 2010

A246431 Smallest m such that A101403(m) = n.

Original entry on oeis.org

0, 2, 1, 3, 727
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 28 2014

Keywords

Comments

A101403(a(n)) = n;
No more terms based on first 10^6 terms of A101402.
No more terms based on the first 3*10^7 terms of A101402. - Robert G. Wilson v, Sep 19 2014

Examples

			a(5) = A129117(1) = 727.

Crossrefs

Cf. A101403, A101402.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a246431 = fromJust . (`elemIndex` a101403_list)

A248497 Decimal expansion of constant mentioned in A247562.

Original entry on oeis.org

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

Views

Author

Odimar Fabeny, Oct 07 2014

Keywords

Examples

			c = 0.3549412150654756947265447261312365125932387...

Crossrefs

Cf. A101402, A247562.
Showing 1-8 of 8 results.

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