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.

A160144 Numerator of (2*n+1)/(2^(2*n+1)-1).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 3, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 9, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 15, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127
Offset: 0

Views

Author

Peter Luschny, May 03 2009

Keywords

Comments

This first differs from A005408 (the odd numbers 2n+1) at a(10). The sequence of differences is A160145. This explains the similarity of A009843 (expansion of x/cos(x)) and A160143. A156769 describes a similar companion to A036279 (expansion of tan(x)).

Links

Crossrefs

Cf. A005408, A009843, A036279, A156769, A160143, A160145, A303449 (denominators).

Programs

  • Magma
    [Numerator((2*n+1)/(2^(2*n+1)-1)): n in [0..70]]; // Vincenzo Librandi, Apr 25 2018
  • Maple
    seq(numer((2*n+1)/(4^(2*n+1)-2^(2*n+1))),n=0..32);
seq(numer((2*n+1)/(2^(2*n+1)-1)),n=0..50); # Altug Alkan, Apr 21 2018
  • Mathematica
    Array[Numerator[(2 # + 1)/(2^(2 # + 1) - 1)] &, 64, 0] (* Michael De Vlieger, Apr 21 2018 *)
  • PARI
    vector(80,n, n--; numerator((2*n+1)/(4^(2*n+1)-2^(2*n+1)))) \\ Michel Marcus, Jan 31 2015
  • PARI
    forstep(k=1, 1e3, 2, print1(numerator(k/(2^k-1)), ", ")); \\ Altug Alkan, Apr 21 2018

Extensions

More terms from Michel Marcus, Jan 31 2015
Name simplified by Altug Alkan, Apr 21 2018
Further edited by N. J. A. Sloane, Apr 24 2018

A160145 a(n) = the odd number 2n+1 minus the numerator of (2n+1)/(2^(2n+1)-1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 150
Offset: 0

Views

Author

Peter Luschny, May 03 2009

Keywords

Comments

Explains the similarity of the sequences A009843 and A160143. (Cf. also the pair A036279 and A156769.) The first nonzero values occur at n = 10, 31, 52 and 73.
Previous name was: Odd numbers 2n+1 minus the numerators of (2n+1)/(4^(2n+1)-2^(2n+1)), (A005408 - A160144). - Altug Alkan, Apr 21 2018

Links

Crossrefs

Cf. A005408, A009843, A160143, A160144.

Programs

  • Maple
    seq((2*n+1)-numer((2*n+1)/(4^(2*n+1)-2^(2*n+1))),n=0..77);
seq((2*n+1)-numer((2*n+1)/(2^(2*n+1)-1)),n=0..100); # Altug Alkan, Apr 21 2018
  • Mathematica
    Array[# - Numerator[#/(2^# - 1)] &[2 # + 1] &, 78, 0] (* Michael De Vlieger, Apr 21 2018 *)
  • PARI
    forstep(k=1, 1e2, 2, print1(k - numerator(k/(2^k-1)), ", ")); \\ Altug Alkan, Apr 21 2018

Formula

a(n) = A005408(n) - A160144(n).

Extensions

Name simplified by Altug Alkan, Apr 21 2018

A193476 The denominators of the Bernoulli secant numbers at odd indices.

Original entry on oeis.org

2, 56, 992, 16256, 261632, 4192256, 67100672, 1073709056, 17179738112, 274877382656, 628292059136, 70368735789056, 1125899873288192, 18014398375264256, 288230375614840832, 4611686016279904256, 73786976286248271872, 1180591620683051565056
Offset: 0

Views

Author

Peter Luschny, Aug 17 2011

Keywords

Comments

Denominator of the coefficient [x^(2n)] of sec(x)*(2*n+1)!/(4*16^n-2*4^n), that is, a(n) is the denominator of A000364(n)*(2*n+1)/(4*16^n-2*4^n). [Edited by Altug Alkan, Apr 22 2018]
Numerators are A160143. [Corrected by Peter Luschny, Mar 18 2021]
A193475(n) = 4*16^n-2*4^n is similar, but differs at n = 10, 31, 52, 73, 77, 94, ...

Links

Crossrefs

Cf. A000364, A160143, A160144, A193475.

Programs

  • Maple
    gf := (f,n) -> coeff(series(f(x),x,n+4),x,n):
A193476 := n -> denom(gf(sec,2*n)*(2*n+1)!/(4*16^n - 2*4^n)):
seq(A193476(n), n = 0..17); # Altug Alkan, Apr 23 2018
  • Mathematica
    a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]*(2j - m)^(2n), {j, 0, m/2}]*(-1)^(k-m), {m, 0, k}], {k, 1, 2n}] (2n+1)/ (4*16^n - 2*4^n) // Denominator; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 26 2019, after Vladimir Kruchinin in A000364 *)
  • PARI
    a(n) = denominator(subst(bernpol(2*n+1), 'x, 1/4)*2^(2*n+1)/(2^(2*n+1)-1)); \\ Altug Alkan, Apr 22 2018 after Charles R Greathouse IV at A000364
Showing 1-3 of 3 results.

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

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