A026255 -id:A026255 - cp's OEIS frontend

A026257 a(n) = (1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026255.

1, 2, 4, 5, 3, 6, 11, 12, 7, 8, 9, 17, 18, 10, 19, 24, 13, 25, 14, 15, 16, 30, 31, 32, 37, 20, 38, 21, 22, 23, 43, 44, 45, 26, 49, 50, 27, 51, 28, 29, 56, 57, 58, 33, 62, 63, 34, 64, 35, 36, 69, 70, 71, 39, 40, 75, 76, 41, 77, 42, 82, 83, 84

Offset: 1

Clark Kimberling

nonn

A026259 a(n) = (1/2)*s(n), where s(n) is the n-th even number in A026255.

Original entry on oeis.org

1, 2, 3, 7, 4, 8, 9, 5, 6, 13, 14, 15, 10, 20, 11, 21, 22, 12, 26, 27, 28, 16, 17, 33, 18, 34, 35, 19, 39, 40, 41, 23, 24, 46, 25, 47, 52, 53, 54, 29, 30, 31, 59, 32, 60, 65, 66, 67, 36, 37, 38, 72, 73, 78, 42, 79, 80, 43, 44, 45, 85, 86, 48

Offset: 1

Clark Kimberling

nonn

A193621 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(x)^A026255(n).

Original entry on oeis.org

1, 1, 3, 9, 32, 122, 490, 2044, 8769, 38455, 171606, 776763, 3557681, 16457402, 76778667, 360830164, 1706641162, 8117569255, 38804142203, 186323145806, 898247214881, 4346078073871, 21097315227638, 102721050351404, 501515949459113, 2454747530072567, 12043165949629976

Offset: 0

Paul D. Hanna, Sep 01 2011

nonn

Sequence A026255 is a self-inverse permutation of the natural numbers where
A026255([k*sqrt(3)]) = [k*(3+sqrt(3))/2] and
A026255([k*(3+sqrt(3))/2]) = [k*sqrt(3)] for k>=1, and [x] = floor(x).

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 122*x^5 + 490*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^4 + x^4*A(x)^3 + x^5*A(x)^7 + x^6*A(x)^9 + x^7*A(x)^5 + x^8*A(x)^11 + x^9*A(x)^6 + x^10*A(x)^14 +...
which also equals: A(x) = 1 + A(x)*x^2 + A(x)^2*x + A(x)^3*x^4 + A(x)^4*x^3 + A(x)^5*x^7 + A(x)^6*x^9 + A(x)^7*x^5 + A(x)^8*x^11 + A(x)^9*x^6 + A(x)^10*x^14 +...
In the above series, the exponents begin:
A026255 = [2,1,4,3,7,9,5,11,6,14,8,16,18,10,21,12,23,13,26,28,15,30...].

Cf. A193620.

{a(n)=local(A=1+x,s=sqrt(3),t=(3+sqrt(3))/2);for(i=1,n,A=1+sum(m=1, n, x^floor(m*s)*(A+x*O(x^n))^floor(m*t)+ x^floor(m*t)*(A+x*O(x^n))^floor(m*s))); polcoeff(A, n)}

G.f. satisfies: A(x) = 1 + Sum_{n>=1} A(x)^n * x^A026255(n).

A026256 a(n) = |s(n) - n|, where s = A026255.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 3, 4, 3, 4, 5, 4, 6, 4, 6, 5, 7, 8, 6, 8, 6, 9, 10, 7, 10, 8, 11, 8, 11, 12, 9, 13, 10, 13, 10, 14, 15, 11, 15, 11, 16, 12, 16, 17, 13, 18, 13, 18, 19, 14, 20, 15, 20, 15, 21, 22, 16, 22, 16, 23, 17, 23, 24, 18, 25, 18, 25, 19

Offset: 1

Clark Kimberling

nonn

A022838 Beatty sequence for sqrt(3); complement of A054406.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112

Offset: 1

Clark Kimberling

nonn

0 <= A144077(n) - a(n) <= 1. - Reinhard Zumkeller, Sep 09 2008
From Reinhard Zumkeller, Jan 20 2010: (Start)
A080757(n) = a(n+1) - a(n).
A171970(n) = floor(a(n)/2).
A171972(n) = a(A000290(n)). (End)
Numbers k>0 such that A194979(k+1) = A194979(k) + 1. - Clark Kimberling, Dec 02 2014
Powers of 2 (i.e, 1, 8, 32, 64, 128, 256, 512, 4096, 8192,...) appear at n=1, 5, 19, 37, 74, 148, 296, 2365, 4730, 18919, 75675, 151349, 302698, 605396, ... related to A293328. - R. J. Mathar, Jan 17 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A080757 (first differences), A194106 (partial sums), A194028 (even bisection), A184796 (prime terms).

Cf. A026255, A054406 (complement).

a022838 = floor . (* sqrt 3) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

[Floor(n*Sqrt(3)): n in [1..60]]; // G. C. Greubel, Sep 28 2018

A022838 := proc(n)
    floor(n*sqrt(3)) ;
end proc: # R. J. Mathar, Mar 25 2013

Table[Floor[n 3^(1/2)] , {n, 1, 65}] (* Geoffrey Critzer, Jan 11 2015 *)

vector(60, n, floor(n*sqrt(3))) \\ G. C. Greubel, Sep 28 2018

a(n)=sqrtint(3*n^2) \\ Charles R Greathouse IV, Nov 01 2021

from math import isqrt
def A022838(n): return isqrt(3*n*n) # Chai Wah Wu, Aug 06 2022

a(n) = floor(n*sqrt(3)). - Reinhard Zumkeller, Jan 20 2010

a(n) = 2 * floor(n * (sqrt(3) - 1)) + floor(n * (2 - sqrt(3))) + 1. - Miko Labalan, Dec 03 2016

A054406 Beatty sequence for (3+sqrt 3)/2; complement of A022838.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 56, 59, 61, 63, 66, 68, 70, 73, 75, 78, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141, 144, 146

Offset: 1

Eric W. Weisstein

nonn

Numbers k such that A194979(k+1) = A194979(k). - Clark Kimberling, Dec 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A194143 (partial sums), A182778 (even bisection), A184799 (prime terms).
Cf. A022838 (complement), A026255.
Cf. A194979.

[Floor(n*(3+Sqrt(3))/2): n in [1..70]]; // Vincenzo Librandi, Oct 25 2011

A054406 := proc(n) n*(3+sqrt(3))/2 ; floor(%) ; end proc: # R. J. Mathar, Feb 26 2011

a054406[n_Integer] := Floor[# (3 + Sqrt[3])/2] & /@ Range[n]; a054406[62] (* Michael De Vlieger, Dec 14 2014 *)

is(n)=sqrtint((n+1)^2\3)==sqrtint(n^2\3) \\ Charles R Greathouse IV, Nov 01 2021

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

A026262 Beginning with the natural numbers, swap [ ksqrt(5) ] and [ (k/4)(5 + sqrt(5)) ], for all k >= 1.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 11, 13, 9, 15, 10, 17, 12, 20, 14, 22, 24, 16, 26, 18, 29, 19, 31, 21, 33, 35, 23, 38, 25, 40, 27, 42, 28, 44, 46, 30, 49, 32, 51, 34, 53, 36, 55, 37, 58, 60, 39, 62, 41, 64, 43, 67, 45, 69, 71, 47, 73, 48, 76, 50, 78

Offset: 1

Clark Kimberling

nonn

Self-inverse integer permutation induced by Beatty sequences for sqrt(5) and (5+sqrt(5))/4. - 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, A026255.

a(A022839(n))=A108598(n) and a(A108598(n))=A022839(n). - Reinhard Zumkeller, Jun 11 2005

cp's OEIS Frontend

A026257 a(n) = (1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026255.

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A026259 a(n) = (1/2)*s(n), where s(n) is the n-th even number in A026255.

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A193621 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(x)^A026255(n).

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A026256 a(n) = |s(n) - n|, where s = A026255.

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A022838 Beatty sequence for sqrt(3); complement of A054406.

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A054406 Beatty sequence for (3+sqrt 3)/2; complement of A022838.

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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

A026262 Beginning with the natural numbers, swap [ k*sqrt(5) ] and [ (k/4)*(5 + sqrt(5)) ], for all k >= 1.

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Author

Keywords

Comments

Links

Crossrefs

Formula

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

A026262 Beginning with the natural numbers, swap [ ksqrt(5) ] and [ (k/4)(5 + sqrt(5)) ], for all k >= 1.