A092259 -id:A092259 - cp's OEIS frontend

A264605 Number of self-dual negacyclic codes of length 4n over GF(3), where 4n runs through the numbers congruent to 4 or 8 mod 12 (cf. A092259).

Original entry on oeis.org

2, 2, 2, 8, 8, 2, 32, 8, 32, 8, 2, 8, 8, 32, 8, 8, 32, 32, 8, 8, 8, 2, 32, 512, 32, 8, 32, 2048, 8, 8, 8, 8, 32, 512

Offset: 1

N. J. A. Sloane, Nov 24 2015

Amita Sahni and Poonam Trama Sehgal, Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields, Advances in Mathematics of Communications, Volume 9, No. 4, 2015, 437-447.

Cf. A092259.

A092260 Permutation of natural numbers generated by 6-rowed array shown below.

Original entry on oeis.org

1, 11, 2, 13, 10, 3, 23, 14, 9, 4, 25, 22, 15, 8, 5, 35, 26, 21, 16, 7, 6, 37, 34, 27, 20, 17, 12, 47, 38, 33, 28, 19, 18, 49, 46, 39, 32, 29, 24, 59, 50, 45, 40, 31, 30, 61, 58, 51, 44, 41, 36, 71, 62, 57, 52, 43, 42, 73, 70, 63, 56, 53, 48, 83, 74, 69, 64, 55, 54, 85, 82, 75

Offset: 1

Giovanni Teofilatto, Feb 19 2004

nonn

11 13 23 25 35 37 47 49 59... (A091998)
10 14 22 26 34 38 46 50 58... (A091999)
9 15 21 27 33 39 45 51 57... (A016945)
8 16 20 28 32 40 44 52 56... (A092259)
7 17 19 29 31 41 43 53 55... (A092242)
12 18 24 30 36 42 48 54 60... (A008588, excluding initial term)
For such arrays A_k, here A_6, see a W. Lang comment on A113807, the A_7 case. However, to get the array A_6 one should take the last line as the first one and add a 0 in front (thus obtaining a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A090298, A113807.

Edited and extended by Ray Chandler, Feb 21 2004

A157697 Decimal expansion of sqrt(2/3).

Original entry on oeis.org

8, 1, 6, 4, 9, 6, 5, 8, 0, 9, 2, 7, 7, 2, 6, 0, 3, 2, 7, 3, 2, 4, 2, 8, 0, 2, 4, 9, 0, 1, 9, 6, 3, 7, 9, 7, 3, 2, 1, 9, 8, 2, 4, 9, 3, 5, 5, 2, 2, 2, 3, 3, 7, 6, 1, 4, 4, 2, 3, 0, 8, 5, 5, 7, 5, 0, 3, 2, 0, 1, 2, 5, 8, 1, 9, 1, 0, 5, 0, 0, 8, 8, 4, 6, 6, 1, 9, 8, 1, 1, 0, 3, 4, 8, 8, 0, 0, 7, 8, 2, 7, 2, 8, 6, 4

Offset: 0

R. J. Mathar, Mar 04 2009

Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - Amiram Eldar, Mar 06 2022
The standard deviation of a roll of a 3-sided die. - Mohammed Yaseen, Feb 23 2023

			0.81649658092772603273242802490196379732198249355222...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jisho Kotani, Problem 2053, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; Solution to Problem 2053, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
Index entries for algebraic numbers, degree 2.

Cf. A002193, A020763, A092259, A115754, A145439.

Sqrt(2/3); // G. C. Greubel, Mar 30 2018

Maple
```
evalf(sqrt(2/3)) ;
```

RealDigits[Sqrt[2/3], 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

sqrt(2/3) \\ G. C. Greubel, Mar 30 2018

Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
From Michal Paulovic, Dec 08 2022: (Start)
Equals 2 * A020763.
Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
Equals exp(-arctanh(1/5)). - Amiram Eldar, Jul 10 2023
Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

A119315 Numbers with composite numbers as third divisors.

Original entry on oeis.org

4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 49, 52, 56, 64, 68, 76, 80, 81, 88, 92, 99, 100, 104, 112, 116, 117, 121, 124, 125, 128, 136, 140, 148, 152, 153, 160, 164, 169, 171, 172, 176, 184, 188, 196, 200, 207, 208, 212, 220, 224, 232, 236, 243, 244, 248, 256, 260

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A067029(m) > 1 and (A001221(m) = 1 or A020639(m)^2 <= A119288(m)).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 23, 221, 2194, 21895, 219307, 2193435, 21937419, 219396872, 2193979781, ... . Apparently, the asymptotic density of this sequence exists and equals 0.219... . - Amiram Eldar, Jul 02 2022
Numbers k such that A292269(k) is composite, which must then be a square of prime (A001248) by necessity. - Antti Karttunen, Jul 02 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A119316.
A025475, A092259, and A355445 are subsequences.
Cf. A000005, A001221, A001248, A002808, A020639, A027750, A067029, A292269, A355453 (characteristic function).
Cf. also A355455.

Select[Range[300],CompositeQ[Divisors[#][[3]]]&]//Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 03 2021 *)
Select[Range[260], (f = FactorInteger[#])[[1, 2]] > 1 && (Length[f] == 1 || f[[1, 1]]^2 < f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

A355453(n) = ((n>1) && !isprime(n) && !isprime(divisors(n)[3]));
isA119315(n) = A355453(n); \\ Antti Karttunen, Jul 02 2022

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

Original entry on oeis.org

1, 1, 1, 5, 3, 5, 5, 0, 7, 1, 6, 5, 0, 4, 1, 0, 5, 4, 0, 7, 6, 7, 0, 5, 8, 3, 7, 4, 5, 5, 5, 8, 3, 0, 9, 3, 7, 9, 4, 5, 8, 2, 7, 1, 8, 4, 4, 6, 4, 5, 8, 5, 7, 2, 4, 6, 6, 0, 4, 5, 5, 2, 9, 6, 8, 7, 0, 5, 2, 6, 3, 0, 2, 1, 4, 0, 6, 0, 6, 0, 2, 3, 8, 4, 8, 5, 0, 3, 6, 7, 2, 6, 8

Offset: 1

R. J. Mathar, Feb 08 2009

			1.11535507165041054076705837455583093794582718446458...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, 1996, 4.1.49.

Cf. A010503, A020760, A020763, A092259.

Maple
```
1/2*(1+1/3*3^(1/2))*2^(1/2);
```

RealDigits[1/Sqrt[2] + 1/Sqrt[6], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

1/sqrt(6) + 1/sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals (1+A020760)*A010503.
Equals A020763 + A010503. - Artur Jasinski, Dec 20 2020
The minimal polynomial is 9*x^4 - 12*x^2 + 1. - Joerg Arndt, Sep 20 2023
Equals 2F1(1/4,3/4; 1/2; 1/4). - R. J. Mathar, Aug 02 2024
Equals Product_{k>=1} (1 - (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

Typo in definition corrected by R. J. Mathar, Feb 09 2009

A119314 Complement of A119313.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 47, 49, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 76, 79, 80, 81, 83, 88, 89, 92, 97, 99, 100, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 136, 137

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) <= 1 or (A067029(m) > 1 and A020639(m)^2 <= A119288(m)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A119315 and A008578.
(Intersection with A119316) = A008578.
A000961 and A092259 are subsequences.
Cf. A001221, A020639, A067029, A119288, A119313.

Select[Range[140], !CompositeQ[#] || ((f = FactorInteger[#])[[1, 2]] > 1 && (Length[f] == 1 || f[[1, 1]]^2 < f[[2, 1]])) &] (* Amiram Eldar, Jul 02 2022 *)

A174398 Numbers that are congruent to {1, 4, 5, 8} mod 12.

Original entry on oeis.org

1, 4, 5, 8, 13, 16, 17, 20, 25, 28, 29, 32, 37, 40, 41, 44, 49, 52, 53, 56, 61, 64, 65, 68, 73, 76, 77, 80, 85, 88, 89, 92, 97, 100, 101, 104, 109, 112, 113, 116, 121, 124, 125, 128, 133, 136, 137, 140, 145, 148, 149, 152, 157, 160, 161, 164, 169, 172, 173

Offset: 1

Gary Detlefs, Mar 18 2010

Numbers k such that k*(k + 3)/4 + (k + 1)*(k + 2)/6 or k*(5*k + 3)/12 + 1/3 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A087445, A092259.

[n : n in [0..200] | n mod 12 in [1, 4, 5, 8]]; // Wesley Ivan Hurt, Jun 07 2016

Maple
```
seq(3*n +(-1)^floor(n/2), n=0..50);
```

Table[(1+I)*(3*(n-n*I+I-1)+I^(1-n)-I^n)/2, {n, 60}] (* Wesley Ivan Hurt, Jun 07 2016 *)
Select[Range[200],MemberQ[{1,4,5,8},Mod[#,12]]&] (* or *) LinearRecurrence[ {2,-2,2,-1},{1,4,5,8},60] (* Harvey P. Dale, Aug 02 2020 *)

a(n) = 3*n - 3 + (-1)^floor((n-1)/2).
From Wesley Ivan Hurt, Jun 07 2016: (Start)
G.f.: x*(1 + 2*x - x^2 + 4*x^3)/((1 - x)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1 + i)*(3*(n - n*i + i - 1) + i^(1-n) - i^n)/2, where i=sqrt(-1).
a(2*k) = A092259(k), a(2*k-1) = A087445(k). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 + log(2)/2. - Amiram Eldar, Dec 31 2021

A306199 Numbers k having the property that tau(4k) < tau(3k) where tau = A000005.

Original entry on oeis.org

4, 8, 16, 20, 28, 32, 40, 44, 48, 52, 56, 64, 68, 76, 80, 88, 92, 96, 100, 104, 112, 116, 124, 128, 136, 140, 148, 152, 160, 164, 172, 176, 184, 188, 192, 196, 200, 208, 212, 220, 224, 232, 236, 240

Offset: 1

Gary Detlefs, Jan 28 2019

nonn

All terms are divisible by 4.
A092259 (numbers congruent to {4,8} (mod 12)) is a subset.
Sequence also includes all numbers of the form 48*k where k is congruent to {1,2} (mod 3) (A001651).
Additional entries of the form 48k, where k is divisible by three have k values of 12*{1,2,4,5,7,8,10,11,12,13,14,16,17,19,20,22,23,24,...}
From Robert Israel, Jan 29 2019: (Start)
Numbers k such that A007814(k)- 2*A007949(k) >= 2.
Sequence is closed under multiplication. (End)
The asymptotic density of this sequence is 2/11. - Amiram Eldar, Mar 25 2021

			tau(4*20) = 10, tau(3*20)=12. So 20 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001651, A007814, A007949, A092259.

with(numtheory): for n from 1 to 300 do if tau(4*n) < tau(3*n) then print(n) fi od

Select[Range[4, 240, 4], #1 < #2 & @@ DivisorSigma[0, # {4, 3}] &] (* Michael De Vlieger, Jan 29 2019 *)
Select[Range[240], IntegerExponent[#, 2] - 2 * IntegerExponent[#, 3] >= 2 &] (* Amiram Eldar, Mar 25 2021 *)

cp's OEIS Frontend

A264605 Number of self-dual negacyclic codes of length 4n over GF(3), where 4n runs through the numbers congruent to 4 or 8 mod 12 (cf. A092259).

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A092260 Permutation of natural numbers generated by 6-rowed array shown below.

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Extensions

A157697 Decimal expansion of sqrt(2/3).

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A119315 Numbers with composite numbers as third divisors.

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A145439 Decimal expansion of Sum_{k>=0} binomial(4*k, 2*k)/2^(6*k).

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Maple

Mathematica

PARI

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A119314 Complement of A119313.

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Mathematica

A174398 Numbers that are congruent to {1, 4, 5, 8} mod 12.

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A306199 Numbers k having the property that tau(4*k) < tau(3*k) where tau = A000005.

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A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

A306199 Numbers k having the property that tau(4k) < tau(3k) where tau = A000005.