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 1-3 of 3 results.

A061037 Numerator of 1/4 - 1/n^2.

Original entry on oeis.org

0, 5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165, 12, 221, 63, 285, 20, 357, 99, 437, 30, 525, 143, 621, 42, 725, 195, 837, 56, 957, 255, 1085, 72, 1221, 323, 1365, 90, 1517, 399, 1677, 110, 1845, 483, 2021, 132, 2205, 575, 2397, 156, 2597, 675
Offset: 2

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Author

N. J. A. Sloane, May 26 2001

Keywords

Comments

From Balmer spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
a(-2) = 0, a(-1) = a(1) = -3. - Paul Curtz, Feb 19 2011
Can be thought of as 4 interlocking sequences, each of the form a(n) = 3a(n - 1) - 3a(n - 2) + a(n - 3). - Charles R Greathouse IV, May 27 2011

References

  • J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

Links

Crossrefs

Cf. A061038 (denominators), A061035-A061050, A126252, A028347.

Programs

  • Haskell
    import Data.Ratio ((%), numerator)
a061037 n = numerator (1%4 - 1%n^2)  -- Reinhard Zumkeller, Dec 17 2011
  • Magma
    [ Numerator(1/4-1/n^2): n in [2..52] ]; // Bruno Berselli, Feb 10 2011
  • Mathematica
    f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 4] &, 51, 0]
f[n_] := Numerator[(n - 2) (n + 2)/(4 n^2)]; Array[f, 51, 2] (* Or *)
a[n_] := 3 a[n - 4] - 3 a[n - 8] + a[n - 12]; a[1] = -3; a[2] = 0; a[3] = 5; a[4] = 3; a[5] = 21; a[6] = 2; a[7] = 45; a[8] = 15; a[9] = 77; a[10] = 6; a[11] = 117; a[12] = 35; Array[a, 51, 2] (* Robert G. Wilson v *)
Numerator[1/4-1/Range[2,60]^2] (* Harvey P. Dale, Aug 18 2011 *)
  • PARI
    a(n) = { numerator(1/4 - 1/n^2) } \\ Harry J. Smith, Jul 17 2009

Formula

G.f.: x^2(-3x^11-x^10-3x^9+14x^7+6x^6+30x^5+2x^4+21x^3+3x^2+5x)/(1-x^4)^3.
a(4n+2) = n(n+1), a(2n+3) = (2n+1)(2n+5), a(4n+4) = (2n+1)(2n+3). - Ralf Stephan, Jun 10 2005
a(n+2) = A060819(n) * A060819(n+4).
a(n) = (n^2-4)*(3*i^n+3*(-i)^n-27*(-1)^n+37)/64, where i is the imaginary unit. - Bruno Berselli, Feb 10 2011
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). - Paul Curtz, Feb 28 2011
a(n+2) = n*(n+4)/(period 4: 16, 1, 4, 1 = A146160(n)) = A028347(n+2) / A146160(n). - Paul Curtz, Mar 24 2011 [edited by Franklin T. Adams-Watters, Mar 25 2011]
a(n) = (n^2-4) / gcd(4*n^2, (n^2-4)). - Colin Barker, Jan 13 2014
Sum_{n>=3} 1/a(n) = 11/6. - Amiram Eldar, Aug 12 2022

A188134 a(4*n) = n, a(1+2*n) = 4+8*n, a(2+4*n) = 2+4*n.

Original entry on oeis.org

0, 4, 2, 12, 1, 20, 6, 28, 2, 36, 10, 44, 3, 52, 14, 60, 4, 68, 18, 76, 5, 84, 22, 92, 6, 100, 26, 108, 7, 116, 30, 124, 8, 132, 34, 140, 9, 148, 38, 156, 10, 164, 42, 172, 11, 180, 46, 188, 12, 196, 50, 204, 13, 212, 54, 220, 14, 228, 58, 236, 15, 244, 62
Offset: 0

Views

Author

Paul Curtz, Mar 21 2011

Keywords

Links

Crossrefs

Cf. A016825, A017113, A060819, A061037, A064680, A146160, A176895, A177499.

Programs

  • Magma
    [(64-3*(1+(-1)^n)*(9+(-1)^(n div 2)))*n/16 : n in [0..80]]; // Wesley Ivan Hurt, Jul 06 2016
  • Maple
    A188134:=n->8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)): seq(A188134(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2016
  • Mathematica
    Table[8 n/(11 + 9 Cos[Pi*n] + 12 Cos[n*Pi/2]), {n, 0, 80}] (* Wesley Ivan Hurt, Jul 06 2016 *)
CoefficientList[Series[x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2, {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,2,12,1,20,6,28},70] (* Harvey P. Dale, Aug 14 2019 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(4+2*x+12*x^2+x^3+12*x^4+ 2*x^5 +4*x^6)/(1-x^4)^2)) \\ G. C. Greubel, Sep 20 2018

Formula

a(n) = 2*a(n-4) - a(n-8) for n>7.
a(n) = A176895(n) * A060819(n).
a(n) = (4*A061037(n+2))/(n+4).
a(n) = 4*n / A146160(n).
a(2*n) = A064680(n).
a(1+2*n) = A017113(n).
a(4*n) = a(-4+4*n) + 1.
a(1+4*n) = a(-3+4*n) + 16.
a(2+4*n) = a(-2+4*n) + 4.
a(3+4*n) = a(-1+4*n) + 16. See A177499.
From Bruno Berselli, Mar 22 2011: (Start)
G.f.: x*(4+2*x+12*x^2+x^3+12*x^4+2*x^5+4*x^6)/(1-x^4)^2.
a(n) = (64-3*(1+(-1)^n)*(9+i^n))*n/16 with i=sqrt(-1).
a(n)/a(n-4) = n/(n-4) for n>4. (End)
a(n) = 8*n/(11 + 9*cos(Pi*n) + 12*cos(n*Pi/2)). - Wesley Ivan Hurt, Jul 06 2016
a(n) = lcm(4,n)/gcd(4,n). - R. J. Mathar, Feb 12 2019
Sum_{k=1..n} a(k) ~ (37/32)*n^2. - Amiram Eldar, Oct 07 2023

A145996 Numbers k such that quintic polynomial 4*k - k^2 + 5*k^2*x + (20*k - 20*k^2)*x^3 + (16 - 32*k + 16*k^2)*x^5 has a rational root.

Original entry on oeis.org

0, 1, 2, 4, 243
Offset: 1

Views

Author

Artur Jasinski, Oct 26 2008

Keywords

Comments

When k = 1 the polynomial degenerates to degree 1.
Conjecture: This sequence is finite and complete.
This sequence is not the same as A005275 because 198815685282 does not belong to this sequence.
No more values of k less than 2*10^7.
One of the root of quintic polynomial 4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5 is Hypergeometric2F1(1/5,4/5,1/2,1/k).
Precisely for k belonging to this sequence, Hypergeometric2F1(1/5,4/5,1/2,1/k) is algebraic number of 4 degree, otherwise it is of degree 5. [Artur Jasinski, Oct 26 2008]
= sqrt(k/(k-1)) cos(3/5 arcsin(1/sqrt(k))). [Artur Jasinski, Oct 29 2008]

Crossrefs

Cf. A146160.

Programs

  • Mathematica
    a = {}; Do[If[Length[FactorList[(4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5)]] > 2, AppendTo[a, k]; Print[k]], {k, 1, 20000000}]; a
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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