A059648 -id:A059648 - cp's OEIS frontend

A059649 Positions of ones in A059648.

2, 7, 9, 12, 14, 19, 24, 26, 31, 36, 38, 41, 43, 48, 50, 53, 55, 60, 65, 67, 70, 72, 77, 79, 82, 84, 89, 94, 96, 101, 106, 108, 111, 113, 118, 123, 125, 130, 135, 137, 140, 142, 147, 149, 152, 154, 159, 164, 166, 171, 176, 178, 181, 183, 188, 193, 195, 200, 205

Offset: 1

Antti Karttunen, Feb 03 2001

nonn

First differences: A059650.

positions := proc(e,ll) local a,k,l,m; l := ll; m := 1; a := []; while(member(e,l[m..nops(l)],'k')) do a := [op(a),(k+m-1)]; m := k+m; od; RETURN(a); end;

Position[With[{k = Sqrt[2]}, Table[Floor[Floor[k^2*j] - k*Floor[k*j]], {j, 0, 300}]], 1] - 1 // Flatten (* Jean-François Alcover, Mar 06 2016 *)

A007069 First column of spectral array W(sqrt 2).

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 12, 15, 16, 19, 21, 22, 25, 26, 29, 31, 33, 35, 36, 39, 41, 43, 45, 46, 49, 50, 53, 55, 57, 59, 60, 63, 65, 67, 69, 70, 73, 74, 77, 79, 80, 83, 84, 87, 89, 91, 93, 94, 97, 98, 101, 103, 104, 107, 108, 111, 113, 115, 117, 118, 121, 123, 125, 127, 128, 131

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A001951, A059648.

[Floor(Sqrt(2)*Floor(Sqrt(2)*n)): n in [1..100]]; // G. C. Greubel, Aug 16 2018

Table[Floor[Sqrt[2]*Floor[Sqrt[2]*n]], {n, 1, 100}] (* G. C. Greubel, Aug 16 2018 *)

vector(100,n, floor(sqrt(2)*floor(n*sqrt(2)))) \\ G. C. Greubel, Aug 16 2018

from math import isqrt
def A007069(n): return isqrt(isqrt(n**2<<1)**2<<1) # Chai Wah Wu, Aug 29 2022

From Benoit Cloitre, Apr 06 2003: (Start)
Iterated floor function: a(n) = floor(sqrt(2)*floor(sqrt(2)*n)).
a(n) = A001951(A001951(n)).
a(n) = 2*n - 1 - A059648(n). (End)

More terms from Benoit Cloitre, Apr 06 2003

A059651 a(n) = [[(k^2)n]-(k[k*n])], where k = cube root of 2 and [] is the floor function.

Original entry on oeis.org

0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1, -1, 0, 1, -1, 0, -1, 0, 0, -1, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 1, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, 0, 0, -1, -1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0

Offset: 0

Antti Karttunen, Feb 03 2001

sign

The values of (floor((k^2)*j)-(k*(floor(k*j)))) for j=0..50, where k=2^(1/3), are 0, -0.259921, 0.480158, 0.220237, -0.299605, -0.559526, 0.180553, 0.92063, -0.59921, ...

A059648 gives similar sequence for k=sqrt(2). Positions of +1's: A059657, positions of -1's A059659.

Digits := 89; floor_diffs_floored(evalf(2^(1/3)),120);

A059652 a(n) = [[(k^2)n]-(k[k*n])], where k = sqrt(3/2) and [] is the floor function.

Original entry on oeis.org

0, -1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0

Offset: 0

Antti Karttunen, Feb 03 2001

sign

The values of (floor((k^2)*j)-(k*(floor(k*j)))) for j=0..45, with k=2^(1/3), are 0, -0.224746, 0.550508, 0.325762, 1.101016, -0.348476, 0.426778, 0.202032, 0.97729, ...

A059648 gives similar sequence for k=sqrt(2). Positions of ones: A059653, positions of minus ones: A059655.

Digits := 89; floor_diffs_floored(sqrt(3/2),120);

A138330 Beatty discrepancy (defined in A138253) giving the closeness of the pair (A136497,A136498) to the Beatty pair (A001951,A001952).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Mar 14 2008

nonn

Old definition was "Beatty discrepancy of the complementary equation b(n) = a(a(n)) + a(n)".

			d(1) - c(c(1)) - c(1) =  3 - 1 - 1 = 1;
d(2) - c(c(2)) - c(2) =  6 - 2 - 2 = 2;
d(3) - c(c(3)) - c(3) = 10 - 5 - 4 = 1;
d(4) - c(c(4)) - c(4) = 13 - 7 - 5 = 1.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Cf. A001951, A001952, A136497, A136498, A138253, A059648.

[2*n - Floor(Sqrt(2)*Floor(Sqrt(2)*n)): n in [1..100]]; // Vincenzo Librandi, Nov 12 2018

a:=n->2*n-floor(sqrt(2)*floor(sqrt(2)*n)): seq(a(n),n=1..120); # Muniru A Asiru, Nov 11 2018

Table[2 n - Floor[Sqrt[2] Floor[Sqrt[2] n]], {n, 1, 100}] (* Vincenzo Librandi, Nov 12 2018 *)

a(n)=2*n-floor(sqrt(2)*floor(sqrt(2)*n)) \\ Benoit Cloitre, May 08 2008

from math import isqrt
def A138330(n): return (m:=n<<1)-isqrt(isqrt(n*m)**2<<1) # Chai Wah Wu, Aug 29 2022

a(n) = d(n) - c(c(n)) - c(n), where c(n) = A001951 and d(n) = A001952.
a(n) = 2*n - A007069(n). - Benoit Cloitre, May 08 2008
a(n) = A059648(n+1) + 1. - Michel Dekking, Nov 11 2018

Definition revised by N. J. A. Sloane, Dec 16 2018

cp's OEIS Frontend

A059649 Positions of ones in A059648.

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A007069 First column of spectral array W(sqrt 2).

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A059651 a(n) = [[(k^2)*n]-(k*[k*n])], where k = cube root of 2 and [] is the floor function.

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Maple

A059652 a(n) = [[(k^2)*n]-(k*[k*n])], where k = sqrt(3/2) and [] is the floor function.

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Author

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Comments

Crossrefs

Programs

Maple

A138330 Beatty discrepancy (defined in A138253) giving the closeness of the pair (A136497,A136498) to the Beatty pair (A001951,A001952).

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Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A059651 a(n) = [[(k^2)n]-(k[k*n])], where k = cube root of 2 and [] is the floor function.

A059652 a(n) = [[(k^2)n]-(k[k*n])], where k = sqrt(3/2) and [] is the floor function.