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 1-3 of 3 results.

A194738 Number of k such that {k*sqrt(3)} < {n*sqrt(3)}, where { } = fractional part.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 1, 7, 5, 3, 1, 10, 7, 4, 15, 11, 7, 3, 17, 12, 7, 2, 19, 13, 7, 1, 21, 14, 7, 29, 21, 13, 5, 30, 21, 12, 3, 31, 21, 11, 1, 32, 21, 10, 43, 31, 19, 7, 43, 30, 17, 4, 43, 29, 15, 56, 41, 26, 11, 55, 39, 23, 7, 54, 37, 20, 3, 53, 35, 17, 69, 50, 31, 12, 67
Offset: 1

Views

Author

Clark Kimberling, Sep 02 2011

Keywords

Comments

Related sequences:
A019587, A194733, A019588, A194734; |r|=(1+sqrt(5))/2
A054072, A194735, A194736, A194737; |r|=sqrt(2)
A194738, A194739, A194740, A194741; |r|=sqrt(3)
A194742, A194743, A194744, A194745; |r|=sqrt(5)
A194746, A194747, A194748, A194749; |r|=sqrt(6)
A194750, A194751, A194752, A194753; |r|=e
A194754, A194755, A194756, A194757; |r|=pi
A194758, A194759, A194760, A194761; |r|=log(2)
A194762, A194763, A194764, A194765; |r|=2^(1/3)
In each case, trivially, the sum of the first two sequences is A000027(for n>0), and likewise for the sum of the other two.

Examples

			{r}=0.7...; {2r}=0.4...; {3r}=0.1...;
{4f}=0.9...; {5r}=0.6...; so that a(5)=3.

Crossrefs

Cf. A194739, A194740.

Programs

  • Mathematica
    r = Sqrt[3]; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194738 *)
Table[t[n], {n, 1, 100}]   (* A194739 *)

A019587 The left budding sequence: number of i such that 0 < i <= n and 0 < {phi*i} <= {phi*n}, where {} denotes the fractional part and phi = A001622.

Original entry on oeis.org

1, 1, 3, 2, 1, 5, 3, 8, 5, 2, 9, 5, 1, 10, 5, 15, 9, 3, 15, 8, 21, 13, 5, 20, 11, 2, 19, 9, 27, 16, 5, 25, 13, 1, 23, 10, 33, 19, 5, 30, 15, 41, 25, 9, 37, 20, 3, 33, 15, 46, 27, 8, 41, 21, 55, 34, 13, 49, 27, 5, 43, 20, 59, 35, 11, 52, 27, 2, 45, 19, 63, 36, 9, 55, 27, 74, 45, 16, 65
Offset: 1

Views

Author

N. J. A. Sloane and J. H. Conway

Keywords

Examples

			{r} = 0.61...; {2r} = 0.23...; {3r} = 0.85...; {4r} = 0.47...; so that a(4) = 2.

References

  • J. H. Conway, personal communication.

Links

Crossrefs

Cf. A001622, A019588, A194733, A193738.

Programs

  • Haskell
    a019587 n = length $ filter (<= nTau) $
            map (snd . properFraction . (* tau) . fromInteger) [1..n]
   where (_, nTau) = properFraction (tau * fromInteger n)
         tau = (1 + sqrt 5) / 2
-- Reinhard Zumkeller, Jan 28 2012
  • Maple
    Digits := 100;
A019587 := proc(n::posint)
    local a,k,phi,kfrac,nfrac ;
    phi := (1+sqrt(5))/2 ;
    a :=0 ;
    nfrac := n*phi-floor(n*phi) ;
    for k from 1 to n do
        kfrac := k*phi-floor(k*phi) ;
        if evalf(kfrac-nfrac)  <= 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A019587(n),n=1..100) ; # R. J. Mathar, Aug 13 2021
  • Mathematica
    r = GoldenRatio; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A019587 *)
Table[t[n], {n, 1, 100}]  (* A194733 *)
(* Clark Kimberling, Sep 02 2011 *)

Formula

a(n) + A194733(n) = n.
The theorem prover Walnut (see link) can compute the following "linear representation" for a(n). Let v = [1,0,0,0,0,0,0,0,0,0,0,0], w = [0,1,1,3,2,1,5,3,8,5,2,9]^T, mu(0) =[[1,0,0,0,0,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0,0,0,0], [0,0,0,1,0,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0,0,0,0], [0,0,0,0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,0,-1,0,0,2,0], [0,0,-2,0,0,1,0,2,0,0,0,0], [0,0,-1,-1,0,1,0,1,1,0,-1,1], [0,0,-1,0,0,0,0,0,0,0,2,0]], mu(1) = [[0,1,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0], [0,0,0,0,0,0,-1,0,0,2,0,0], [0,-2,0,0,1,0,2,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0], [0,-1,0,0,0,0,0,0,0,2,0,0], [0,-4,0,0,2,0,4,0,0,-1,0,0]]. Then a(n) = v.mu(x).w, where x is the Zeckendorf (or Fibonacci) representation of n. This gives an algorithm for a(n) that runs in polynomial time in log n. - Jeffrey Shallit, Aug 09 2025

Extensions

Extended by Ray Chandler, Apr 18 2009

A019588 The right budding sequence: # of i such that 0 < i <= n and {tau*n} <= {tau*i} < 1, where {} is fractional part.

Original entry on oeis.org

1, 2, 1, 3, 5, 2, 5, 1, 5, 9, 3, 8, 13, 5, 11, 2, 9, 16, 5, 13, 1, 10, 19, 5, 15, 25, 9, 20, 3, 15, 27, 8, 21, 34, 13, 27, 5, 20, 35, 11, 27, 2, 19, 36, 9, 27, 45, 16, 35, 5, 25, 45, 13, 34, 1, 23, 45, 10, 33, 56, 19, 43, 5, 30, 55, 15, 41, 67, 25, 52, 9, 37, 65, 20, 49, 3, 33, 63, 15
Offset: 1

Views

Author

N. J. A. Sloane and J. H. Conway

Keywords

Comments

Also, the number of distinct blocks of the Fibonacci word (A003849) containing the maximum possible number of 1's for such a block. - Jeffrey Shallit, Jul 09 2025

References

  • J. H. Conway, personal communication.

Links

Crossrefs

Cf. A003849, A019587, A194734, A194738.

Programs

  • Haskell
    a019588 n = length $ filter (nTau <=) $
            map (snd . properFraction . (* tau) . fromInteger) [1..n]
   where (_, nTau) = properFraction (tau * fromInteger n)
         tau = (1 + sqrt 5) / 2
-- Reinhard Zumkeller, Jan 28 2012
  • Mathematica
    r = -GoldenRatio; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A019588 *)
Table[t[n], {n, 1, 100}]   (* A194734 *)
(* Clark Kimberling, Sep 02 2011 *)
Fold[Join[#1, Range[#1[[#2]], Length[#1] + 1 + Floor[GoldenRatio (#2 + 1)] - Floor[GoldenRatio #2], #2 + 1]] &, {1, 2}, Range[30]] (* Birkas Gyorgy, May 24 2012 *)

Formula

a(n) = A194733(n) + 1.

Extensions

Extended by Ray Chandler, Apr 18 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.