cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A146535 Numerator of (2*n-1)/3.

Original entry on oeis.org

1, 1, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9, 29, 31, 11, 35, 37, 13, 41, 43, 15, 47, 49, 17, 53, 55, 19, 59, 61, 21, 65, 67, 23, 71, 73, 25, 77, 79, 27, 83, 85, 29, 89, 91, 31, 95, 97, 33, 101, 103, 35, 107, 109, 37, 113, 115, 39, 119, 121, 41, 125, 127, 43, 131, 133, 45
Offset: 1

Views

Author

Artur Jasinski, Oct 31 2008

Keywords

Comments

From Jaroslav Krizek, May 28 2010: (Start)
a(n+1) = numerators of antiharmonic mean of the first n positive integers for n >= 1.
See A169609(n-1) - denominators of antiharmonic mean of the first n positive integers for n >= 1. (End)

Examples

			Fractions begin with 1/6, 1/2, 5/6, 7/6, 3/2, 11/6, 13/6, 5/2, 17/6, 19/6, 7/2, 23/6, ...

Links

Crossrefs

Cf. A146306, A146307, A146308, A146309, A146310, A146311, A146312, A146313.
Cf. A061347, A102283.
Trisections: A016921, A005408, A016969. [R. J. Mathar, Nov 21 2008]

Programs

  • Mathematica
    Table[Numerator[(2 n - 1)/6], {n, 1, 100}]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,5,7,3,11},100] (* Harvey P. Dale, Feb 24 2015 *)
  • PARI
    a(n) = numerator((2*n-1)/3); \\ Altug Alkan, Apr 13 2018

Formula

From R. J. Mathar, Nov 21 2008: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x(1+x)(1+5x^2+x^4)/((1-x)^2*(1+x+x^2)^2). (End)
Sum_{k=1..n} a(k) ~ (7/9) * n^2. - Amiram Eldar, Apr 04 2024
a(n) = (2*n - 1)*(7 - A061347(n) +3*A102283(n))/9. - Stefano Spezia, Feb 14 2025

Extensions

Name edited by Altug Alkan, Apr 13 2018

A146307 a(n) = denominator of (n-6)/(2n) = denominator of (n+6)/(2n).

Original entry on oeis.org

2, 1, 2, 4, 10, 1, 14, 8, 6, 5, 22, 4, 26, 7, 10, 16, 34, 3, 38, 20, 14, 11, 46, 8, 50, 13, 18, 28, 58, 5, 62, 32, 22, 17, 70, 12, 74, 19, 26, 40, 82, 7, 86, 44, 30, 23, 94, 16, 98, 25, 34, 52, 106, 9, 110, 56, 38, 29, 118, 20, 122, 31, 42, 64, 130, 11, 134, 68, 46, 35, 142, 24
Offset: 1

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

For numerators see A146306.
General formula:
2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =
Hypergeometric2F1(A146306(n)/a(n), A146306(n+12)/a(n), 1/2, 3/4).
2*cos(2*Pi/n) is a root of a polynomial of degree EulerPhi(n)/2 = A000010(n)/2 = A023022(n).
Records in this sequence are even and are congruent to 2 or 10 mod 12 (see A091999).
Indices where odd numbers occur in this sequence are 4n-2 (see A016825).
Indices where prime numbers occur in this sequence see A146309.
From Robert Israel, Apr 21 2021: (Start)
a(n) = 2*n if n == 1, 5, 7 or 11 (mod 12).
a(n) = n if n == 4 or 8 (mod 12).
a(n) = 2*n/3 if n == 3 or 9 (mod 12).
a(n) = n/2 if n == 2 or 10 (mod 12).
a(n) = n/3 if n == 0 (mod 12).
a(n) = n/6 if n == 6 (mod 12). (End)
Sum_{k=1..n} a(k) ~ (77/144) * n^2. - Amiram Eldar, Apr 04 2024

Links

Crossrefs

Cf. A007310, A051724, A091999, A146306 (numerators), A146308.

Programs

  • Maple
    f:= n -> denom((n-6)/(2*n)):
map(f, [$1..100]); # Robert Israel, Apr 20 2021
  • Mathematica
    Table[Denominator[(n - 6)/(2 n)], {n, 1, 100}]
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1},{2,1,2,4,10,1,14,8,6,5,22,4,26,7,10,16,34,3,38,20,14,11,46,8},80] (* Harvey P. Dale, May 15 2022 *)

A146306 a(n) = numerator of (n-6)/(2n).

Original entry on oeis.org

-5, -1, -1, -1, -1, 0, 1, 1, 1, 1, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 4, 17, 3, 19, 5, 7, 11, 23, 2, 25, 13, 9, 7, 29, 5, 31, 8, 11, 17, 35, 3, 37, 19, 13, 10, 41, 7, 43, 11, 15, 23, 47, 4, 49, 25, 17, 13, 53, 9, 55, 14, 19, 29, 59, 5, 61, 31, 21, 16, 65, 11, 67, 17, 23, 35, 71, 6, 73, 37
Offset: 1

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

For denominators see A146307.
General formula:
2*cos(2*Pi/n) = Hypergeometric2F1((n-6)/(2n), (n+6)/(2n), 1/2, 3/4) =
Hypergeometric2F1(a(n)/A146307(n), a(n+12)/A146307(n), 1/2, 3/4).
2*cos(2*Pi/n) is root of polynomial of degree = EulerPhi(n)/2 = A000010(n)/2 = A023022(n).
Records in this sequence are congruent to 1 or 5 mod 6 (see A007310).
First occurrence n in this sequence see A146308.

Examples

			Fractions begin with -5/2, -1, -1/2, -1/4, -1/10, 0, 1/14, 1/8, 1/6, 1/5, 5/22, 1/4, ...

Links

Crossrefs

Cf. A000010, A007310, A023022, A051724, A146307 (denominators), A146308.

Programs

  • Mathematica
    Table[Numerator[(n - 6)/(2 n)], {n, 1, 100}]

Formula

a(n+5) = A051724(n).
Sum_{k=1..n} a(k) ~ (77/288) * n^2. - Amiram Eldar, Apr 04 2024
From Chai Wah Wu, May 08 2025: (Start)
a(n) = 2*a(n-12) - a(n-24) for n > 24.
G.f.: x*(x^23 + 7*x^22 + 2*x^21 + 3*x^20 + 5*x^19 + 11*x^18 + x^17 + 13*x^16 + 7*x^15 + 5*x^14 + 4*x^13 + 17*x^12 + x^11 + 5*x^10 + x^9 + x^8 + x^7 + x^6 - x^4 - x^3 - x^2 - x - 5)/(x^24 - 2*x^12 + 1). (End)

A146309 a(n) = indices where primes occurred in A146306.

Original entry on oeis.org

1, 3, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 58, 62, 66, 74, 78, 82, 86, 94, 102, 106, 114, 118, 122, 134, 138, 142, 146, 158, 166, 174, 178, 186, 194, 202, 206, 214, 218, 222, 226, 246, 254, 258, 262, 274, 278, 282, 298, 302, 314, 318, 326, 334, 346, 354, 358
Offset: 0

Views

Author

Artur Jasinski, Oct 29 2008

Keywords

Comments

General formula (*Artur Jasinski*):
2 Cos[2*Pi/n] = Hypergeometric2F1[(n-6)/(2n),(n+6)/(2n),1/2,3/4] =
Hypergeometric2F1[A146306(n)/A146307(n),A146306(n+12)/A146307(n),1/2,3/4].
2 Cos[2*Pi/n] is root of polynomial of degree = EulerPhi[n]/2 = A000010(n)/2 = A023022(n).

Crossrefs

A007310, A051724, A146306, A146307, A146308.

Programs

  • Mathematica
    aa = {}; Do[k = Denominator[(n - 6)/(2 n)]; If[PrimeQ[k], AppendTo[aa, n]], {n, 1, 1000}]; aa (*Artur Jasinski*)
Showing 1-4 of 4 results.

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