A026262 -id:A026262 - cp's OEIS frontend

A026264 (1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026262.

1, 2, 3, 4, 6, 7, 5, 8, 9, 15, 10, 16, 11, 17, 18, 12, 13, 14, 25, 26, 27, 28, 19, 20, 21, 22, 34, 23, 35, 36, 24, 37, 44, 29, 45, 30, 46, 47, 31, 32, 33, 53, 54, 55, 56, 38, 39, 40, 41, 63, 42, 64, 65, 43, 66, 72, 73, 48, 74, 49, 75, 50, 51

Offset: 1

Clark Kimberling

nonn

A026266 a(n) = (1/2)*s(n), where s = A026262.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 7, 11, 12, 8, 13, 9, 19, 20, 21, 14, 22, 23, 15, 16, 17, 18, 29, 30, 31, 32, 24, 38, 25, 39, 26, 40, 41, 27, 42, 28, 48, 49, 50, 33, 51, 34, 35, 36, 37, 57, 58, 59, 60, 61, 43, 67, 44, 68, 45, 69, 70, 46, 47, 76, 77, 78

Offset: 1

Clark Kimberling

nonn

A026263 a(n) = |s(n) - n|, where s = A026262.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 6, 5, 6, 5, 6, 7, 6, 8, 6, 8, 6, 8, 7, 8, 9, 8, 10, 8, 10, 8, 10, 8, 10, 9, 11, 12, 10, 12, 10, 12, 10, 13, 10, 13, 14, 11, 14, 12, 15, 12, 15, 12, 15, 16, 13, 16, 13, 17, 14, 17, 14, 17

Offset: 1

Clark Kimberling

nonn

A022839 Beatty sequence for sqrt(5).

Original entry on oeis.org

2, 4, 6, 8, 11, 13, 15, 17, 20, 22, 24, 26, 29, 31, 33, 35, 38, 40, 42, 44, 46, 49, 51, 53, 55, 58, 60, 62, 64, 67, 69, 71, 73, 76, 78, 80, 82, 84, 87, 89, 91, 93, 96, 98, 100, 102, 105, 107, 109, 111, 114, 116, 118, 120, 122, 125, 127, 129, 131, 134, 136

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. L. Graham, S. Lin and C.-S. Lin, Spectra of numbers, Math. Mag. 51 (1978), 174-176.
Index entries for sequences related to Beatty sequences

Cf. A026262; complement of A108598.

[Floor(n*Sqrt(5)): n in [1..60]]; // Vincenzo Librandi, Oct 22 2011

a:=n->floor(n*sqrt(5)): seq(a(n),n=1..70); # Muniru A Asiru, Sep 28 2018

With[{c=Sqrt[5]},Floor[c*Range[70]]] (* Harvey P. Dale, Mar 19 2012 *)

vector(60, n, floor(n*sqrt(5))) \\ G. C. Greubel, Sep 28 2018

a(n)=sqrtint(5*n^2) \\ Charles R Greathouse IV, Jan 24 2022

from math import isqrt
def A022839(n): return isqrt(5*n**2) # Chai Wah Wu, Sep 07 2022

a(n) = floor(n*sqrt(5)). - G. C. Greubel, Sep 28 2018

A108598 a(n) = floor(n*((5+sqrt(5))/4)).

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 95, 97, 99, 101, 103, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Beatty sequence for (5+sqrt(5))/4; complement of A022839.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A022839, A026262.

[Floor(n*(5+Sqrt(5))/4): n in [1..80]]; // G. C. Greubel, Oct 20 2023

With[{c=(5+Sqrt[5])/4},Floor[c Range[80]]] (* Harvey P. Dale, Jul 20 2022 *)

[int(n*sqrt(5)*golden_ratio/2) for n in range(1,81)] # G. C. Greubel, Oct 20 2023

A026250 Beginning with the natural numbers, swap [ ksqrt(2) ] and [ k(2 + sqrt(2)) ], for all k >= 1.

Original entry on oeis.org

3, 6, 1, 10, 13, 2, 17, 20, 23, 4, 27, 30, 5, 34, 37, 40, 7, 44, 47, 8, 51, 54, 9, 58, 61, 64, 11, 68, 71, 12, 75, 78, 81, 14, 85, 88, 15, 92, 95, 16, 99, 102, 105, 18, 109, 112, 19, 116, 119, 122, 21, 126, 129, 22, 133, 136, 139, 24, 143, 146

Offset: 1

Clark Kimberling

nonn

Self-inverse integer permutation induced by Beatty sequences for sqrt(2) and 2+sqrt(2). - Reinhard Zumkeller, Jun 11 2005

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A026255, A026262.

a(A001951(n)) = A001952(n) and a(A001952(n)) = A001951(n). - Reinhard Zumkeller, Jun 11 2005

A026255 Beginning with the natural numbers, swap [ ksqrt(3) ] and [ (k/2)(3 + sqrt(3)) ], for all k >= 1.

Original entry on oeis.org

2, 1, 4, 3, 7, 9, 5, 11, 6, 14, 8, 16, 18, 10, 21, 12, 23, 13, 26, 28, 15, 30, 17, 33, 35, 19, 37, 20, 40, 22, 42, 44, 24, 47, 25, 49, 27, 52, 54, 29, 56, 31, 59, 32, 61, 63, 34, 66, 36, 68, 70, 38, 73, 39, 75, 41, 78, 80, 43, 82, 45, 85, 46

Offset: 1

Clark Kimberling

nonn

Self-inverse integer permutation induced by Beatty sequences for sqrt(3) and (3+sqrt(3))/2. - Reinhard Zumkeller, Jun 11 2005

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A026250, A026262.

a(A022838(n)) = A054406(n) and a(A054406(n)) = A022838(n). - Reinhard Zumkeller, Jun 11 2005

cp's OEIS Frontend

A026264 (1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026262.

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A026266 a(n) = (1/2)*s(n), where s = A026262.

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A026263 a(n) = |s(n) - n|, where s = A026262.

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A022839 Beatty sequence for sqrt(5).

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Magma

Maple

Mathematica

PARI

PARI

Python

Formula

A108598 a(n) = floor(n*((5+sqrt(5))/4)).

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Programs

Magma

Mathematica

SageMath

A026250 Beginning with the natural numbers, swap [ k*sqrt(2) ] and [ k*(2 + sqrt(2)) ], for all k >= 1.

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Author

Keywords

Comments

Links

Crossrefs

Formula

A026255 Beginning with the natural numbers, swap [ k*sqrt(3) ] and [ (k/2)*(3 + sqrt(3)) ], for all k >= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A026250 Beginning with the natural numbers, swap [ ksqrt(2) ] and [ k(2 + sqrt(2)) ], for all k >= 1.

A026255 Beginning with the natural numbers, swap [ ksqrt(3) ] and [ (k/2)(3 + sqrt(3)) ], for all k >= 1.