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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.

User: Steven J. Forsberg

Steven J. Forsberg's wiki page.

Steven J. Forsberg has authored 2 sequences.

A175784 Numerators of k/(10+k)+1 for k = 2*n-1.

Original entry on oeis.org

12, 16, 4, 24, 28, 32, 36, 8, 44, 48, 52, 56, 12, 64, 68, 72, 76, 16, 84, 88, 92, 96, 20, 104, 108, 112, 116, 24, 124, 128, 132, 136, 28, 144, 148, 152, 156, 32, 164, 168, 172, 176, 36, 184, 188, 192, 196, 40, 204, 208, 212, 216, 44, 224, 228
Offset: 1

Views

Author

Steven J. Forsberg, Dec 04 2010

Keywords

Comments

For even k the expression k/(k+10)+1 yields A060791 as denominators, A096431 as numerators. For odd k it yields A096431 as denominators, the present sequence as numerators.
Note that A096431 is denominator of (9*(n^4 - 2n^3 + 2n^2 - n) + 2)/(2*(2*n-1)), equivalently denominator of (3*n^2 - 3*n + 1)*(3*n^2 - 3*n + 2)/(2*n-1), and that A060791 is n/gcd(n,5).

Examples

			n=1: (2*1-1)/(2*1+9)+1 = 1/11+1 = 12/11, hence a(1) = 12;
n=2: (2*2-1)/(2*2+9)+1 = 3/13+1 = 16/13, hence a(2) = 16;
n=3: (2*3-1)/(2*3+9)+1 = 5/15+1 = 1/3+1 = 4/3, hence a(3) = 4;

Crossrefs

Cf. A096431, A060791.

Programs

  • Maple
    A175784 := proc(n) local k ; k := 2*n-1 ; numer(k/(10+k)+1) ; end proc:
seq(A175784(n),n=1..30) ; # R. J. Mathar, Feb 05 2011
  • Mathematica
    Numerator[Table[k/(10 + k) + 1, {k, 1, 100, 2}]]

Formula

a(n) = numerator((2*n-1)/(2*n+9) + 1).
Conjecture: a(n) = 2*a(n-5) - a(n-10) = 4*A060791(n+2) with g.f. -4*x*(-3 - 4*x - x^2 - 6*x^3 - 7*x^4 - 2*x^5 - x^6 + x^8 + 2*x^9) / ( (x-1)^2*(x^4 + x^3 + x^2 + x + 1)^2 ). [R. J. Mathar, Dec 07 2010]

A117465 Denominator of -16/((n+2)*n*(n-2)*(n-4)).

Original entry on oeis.org

9, 0, 15, 0, 105, 24, 945, 120, 3465, 360, 9009, 840, 19305, 1680, 36465, 3024, 62985, 5040, 101745, 7920, 156009, 11880, 229425, 17160, 326025, 24024, 450225, 32760, 606825, 43680, 801009, 57120, 1038345, 73440, 1324785, 93024, 1666665, 116280
Offset: 1

Views

Author

Steven J. Forsberg, Apr 25 2006

Keywords

Comments

I came up with the equation to help analyze the path to stable orbits of the logistic function
f(n+1) = k*n(1-n) for f(n) with n => 9, then f(n)*A072346(n-5) = A072346(n+3).
a(n) is the denominator of f(n). The numerator of f(n) is -1 if n is even, else -16.

Examples

			f(5) = -16/(7*5*3*1) = -16/105, denominator a(5) = 105.
f(6) = -16/(8*6*4*2) = -1/24, denominator a(6) = 24.

Links

Crossrefs

Cf. A052762, A072346.

Programs

  • Maple
    f(n) := n -> (1/((n/4)+(n^2/4)-(n^3/16)-1))/n;
  • Mathematica
    Join[{9,0,15,0},Denominator[Table[-(16/(n (n^3-4 n^2-4 n+16))), {n,5,40}]]]    (* Harvey P. Dale, Nov 06 2011 *)
  • PARI
    Vec(3*x*(10*x^12-50*x^10+105*x^8-160*x^6-8*x^5-40*x^4+10*x^2-3)/((x-1)^5*(x+1)^5) + O(x^100)) \\ Colin Barker, Nov 11 2014

Formula

a(n) = denominator of the reduced -16/(n*(n-2)*(n+2)*(n-4)).
a(2n) = A052762(n+1).
a(n) = 5*a(n-2) -10*a(n-4) +10*a(n-6) -5*a(n-8) +a(n-10) for n>15. - R. J. Mathar, Mar 27 2010
a(n) = -(-17+15*(-1)^n)*(n*(16-4*n-4*n^2+n^3))/32 for n>3. - Colin Barker, Nov 11 2014
G.f.: 3*x*(10*x^12-50*x^10+105*x^8-160*x^6-8*x^5-40*x^4+10*x^2-3) / ((x-1)^5*(x+1)^5). - Colin Barker, Nov 11 2014

Extensions

Clearer definition from R. J. Mathar, Mar 27 2010

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