A049473 -id:A049473 - cp's OEIS frontend

A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, 46, 49, 52, 56, 59, 63, 66, 69, 73, 76, 80, 83, 87, 90, 93, 97, 100, 104, 107, 110, 114, 117, 121, 124, 128, 131, 134, 138, 141, 145, 148, 151, 155, 158, 162, 165, 169, 172, 175, 179, 182, 186, 189, 192, 196, 199

Offset: 0

N. J. A. Sloane

nonn

Winning positions in the 2-Wythoff game, the u-pile in Connell's nomenclature; v-pile numbers in A001953.

Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Complement of A001953. Bisection of A003152.

[Floor((2+Sqrt(2))*(2*n+1)/2): n in [0..70]]; // G. C. Greubel, Dec 20 2019

seq( floor((2+sqrt(2))*(2*n+1)/2), n=0..70); # G. C. Greubel, Dec 20 2019

Table[Floor[(n + 1/2) (2 + Sqrt[2])], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
Complement[Range[300], Table[Floor[Sqrt[2*n*(n + 1)]], {n, 0, 300}]] (* Ralf Steiner, Oct 27 2019 *)

PARI
```
a(n)=floor((n+1/2)*(2+sqrt(2)))
```

[floor((2+sqrt(2))*(2*n+1)/2) for n in (0..70)] # G. C. Greubel, Dec 20 2019

a(n + 1) - a(n) is either 3 or 4. Note the comment regarding some intervals in the complement (A001953). - Ralf Steiner, Oct 27 2019

More terms from Michael Somos, Apr 26 2000

New name from Hugo Pfoertner, Dec 27 2021

A001953 a(n) = floor((n + 1/2) * sqrt(2)).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94, 95

Offset: 0

N. J. A. Sloane

nonn

Let s(n) = zeta(3) - Sum_{k = 1..n} 1/k^3. Conjecture: for n >= 1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1's, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Complement of A001954.

Cf. A000217 (T), A136119, A001108.

[Floor((2*n+1)/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Nov 14 2019

seq( floor((2*n+1)/sqrt(2)), n=0..100); # G. C. Greubel, Nov 14 2019

Table[Floor[(n + 1/2) Sqrt[2]], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

PARI
```
a(n)=floor((n+1/2)*sqrt(2))
```

a(n)={sqrtint(2*n*(n+1))} \\ Andrew Howroyd, Oct 24 2019

[floor((2*n+1)/sqrt(2)) for n in (0..100)] # G. C. Greubel, Nov 14 2019

From Ralf Steiner, Oct 23 2019: (Start)
a(n) = floor(2*sqrt(A000217(n))).
a(n) = A136119(n + 1) - 1.
a(n + 1) - a(n) is in {1,2}.
a(n + 3) - a(n) is in {4,5}. (End)

More terms from Michael Somos, Apr 26 2000.

A104521 Fixed point of the morphism 0->{1}, 1->{1,0,1}.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Joerg Arndt, Apr 20 2005

nonn

A080764 and this sequence contain (arbitrarily?) long common substrings.
Yes, A080764 and this sequence contain arbitrarily long common substrings, since the morphism 0 -> 1, 1 -> 110 of A080764 and the morphism 0 -> 1, 1 -> 101 generate the same language: the second morphism is a rotation of the first (so they are conjugate to each other). - Michel Dekking, Feb 03 2017
Zak Seidov points out (Mar 17 2006) that essentially the same sequence arises from the following process: Start with {0,1}; between each pair of digits, insert their sum written in binary. We get successively:
 {0,1,1}
 {0,1,1,1,0,1}
 {0,1,1,1,0,1,1,0,1,1,0,1,1}
 {0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1}, etc.,
which is the current sequence without the initial 1.
A Sturmian word with slope sqrt(2)/2 and intercept (3-sqrt(2))/2; see formula. - Jeffrey Shallit, Mar 06 2024

			The evolution starting with 0 is:
  0
  1
  101
  1011101
  10111011011011101
  10111011011011101101110110111011011011101

Joerg Arndt, Matters Computational (The Fxtbook), section 38.12.1 "Pell palindromes", p. 759 (fast algorithm to compute a function whose value at x=1/2 gives the constant 0.7321604330... whose binary value is 0.1011101101101...)

Cf. A049473, A080764.

Nest[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0, 1}}] &, 0, 7] (* Robert G. Wilson v, Apr 23 2005 *)
h[n_] := Floor[n / Sqrt[2] + 1/2]; Table[h[n + 1] - h[n], {n, 0, 104}]
(* Peter Luschny, Mar 06 2024 *)

#! /usr/bin/env zsh
function N { local w=$1; for (( i=0; i<7; i+=1 )); do echo $w; w=$(echo $w | S); done }
function S { sed 's/1/1_1/g; s/0/1/g; s/_/0/g;' } # 0->1, 1->101
N "0"
# Joerg Arndt, Apr 24 2005

a(n) = floor((n+2)a + b)-floor((n+1)a+b), where a = sqrt(2)/2, b = (3-sqrt(2))/2. - Jeffrey Shallit, Mar 06 2024

a(n) = round((n+1)/sqrt(2))-round(n/sqrt(2)). - Chai Wah Wu, Mar 07 2024

A091087 a(n) = floor(r*n) + floor(n/r), where r=sqrt(2).

Original entry on oeis.org

0, 1, 3, 6, 7, 10, 12, 13, 16, 18, 21, 22, 24, 27, 28, 31, 33, 36, 37, 39, 42, 43, 46, 48, 49, 52, 54, 57, 58, 61, 63, 64, 67, 69, 72, 73, 75, 78, 79, 82, 84, 85, 88, 90, 93, 94, 97, 99, 100, 103, 105, 108, 109, 111, 114, 115, 118, 120, 123, 124, 126, 129, 130, 133, 135

Offset: 0

Clark Kimberling, Dec 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A049473.

[Floor(n*Sqrt(2)) + Floor(n/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Sep 27 2018

Table[Floor[n*Sqrt[2]] + Floor[n/Sqrt[2]], {n, 0, 100}] (* G. C. Greubel, Sep 27 2018 *)

vector(100, n, n--; floor(n*sqrt(2)) + floor(n/sqrt(2))) \\ G. C. Greubel, Sep 27 2018

A361795 a(n) is the area of the largest rectangle with integer sides that can be drawn inside a circle of diameter n.

Original entry on oeis.org

0, 0, 1, 4, 6, 12, 16, 20, 30, 36, 49, 56, 64, 81, 90, 110, 121, 144, 156, 169, 196, 210, 240, 256, 272, 306, 324, 361, 380, 420, 441, 462, 506, 529, 576, 600, 625, 676, 702, 756, 784, 812, 870, 900, 961, 992, 1056, 1089, 1122, 1190

Offset: 0

John Mason, Mar 24 2023

nonn

Alternatively a(n) is the area of the largest rectangle with integer sides having a diagonal of length <= n.

a(n) = x*x or x*(x+1) for x=floor(n/sqrt(2)).

Cf. A049472, A049473.

a(n)={my(t=sqrtint(n^2\2)); if(2*t*(t + 1) < n^2, t + 1, t)*t} \\ Andrew Howroyd, Mar 24 2023

a(n) = sqrtint(n^2\2) * ((sqrtint(2*n^2)+1)\2) \\ Andrew Howroyd, Mar 24 2023

a(n) = A049472(n) * A049473(n). - Andrew Howroyd, Mar 24 2023

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A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

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A001953 a(n) = floor((n + 1/2) * sqrt(2)).

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A104521 Fixed point of the morphism 0->{1}, 1->{1,0,1}.

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A091087 a(n) = floor(r*n) + floor(n/r), where r=sqrt(2).

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A361795 a(n) is the area of the largest rectangle with integer sides that can be drawn inside a circle of diameter n.

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