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.

A242376 Numerators of b(n) = b(n-1)/2 + 1/(2*n), b(0)=0.

Original entry on oeis.org

0, 1, 1, 5, 1, 4, 13, 151, 16, 83, 73, 1433, 647, 15341, 28211, 10447, 608, 19345, 18181, 651745, 771079, 731957, 2786599, 122289917, 14614772, 140001721, 134354573, 774885169, 745984697, 41711914513, 80530073893, 4825521853483
Offset: 0

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Author

Paul Curtz, May 12 2014

Keywords

Comments

See the denominators in A241519.
b(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ... (Ta0(n) in A241269) is an autosequence of the first kind.

Examples

			0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, 16/105, 83/630, 73/630, ...

Links

Crossrefs

Cf. A241519 (denominators).

Programs

  • Mathematica
    Table[-Re[LerchPhi[2, 1, n + 1]], {n, 0, 20}] // Numerator (* Eric W. Weisstein, Dec 11 2017 *)
-Re[LerchPhi[2, 1, Range[20]]] // Numerator (* Eric W. Weisstein, Dec 11 2017 *)
RecurrenceTable[{b[n] == b[n - 1]/2 + 1/(2 n), b[0] == 0}, b[n], {n, 20}] // Numerator (* Eric W. Weisstein, Dec 11 2017 *)
  • Sage
    def a():
    b = n = 0
    while True:
        yield numerator(b)
        n = n + 1
        b = (b/2 + 1/(2*n)) # Ralf Stephan, May 18 2014

Formula

0 = b(n)*(+b(n+1) - 4*b(n+2) + 4*b(n+3)) + b(n+1)*(-2*b(n+1) + 9*b(n+2) - 10*b(n+3)) + b(n+2)*(-2*b(n+2) + 4*b(n+3)) if n>=0. - Michael Somos, May 26 2014
b(n) = -Re(Phi(2, 1, n + 1)). - Eric W. Weisstein, Dec 11 2017
G.f. for b(n): -log(1-x)/(2*(1-x/2)). - Vladimir Kruchinin, Nov 14 2022

Extensions

a(14)-a(25) from Jean-François Alcover, May 12 2014
Corrected a(22) and a(24), more terms from Ralf Stephan, May 18 2014

A353250 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.

Original entry on oeis.org

1, 1, 4, 24, 48, 480, 960, 13440, 26880, 161280, 322560, 7096320, 14192640, 369008640, 738017280, 295206912, 590413824, 20074070016, 40148140032, 1525629321216, 15256293212160, 30512586424320, 61025172848640, 2807157951037440, 5614315902074880
Offset: 0

Views

Author

Vladimir Reshetnikov, Apr 08 2022

Keywords

Examples

			a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the numerators: 1, 1, 4, 24, 48, ...

Links

Crossrefs

Cf. A353251 (denominators).
Cf. A003149, A136128, A191778 (has many terms in common), A241519, A242376.

Programs

  • Mathematica
    Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 24}] // Numerator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 24}] // Numerator

Formula

a(n) = numerator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.

A353251 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); denominators.

Original entry on oeis.org

1, 1, 3, 13, 19, 143, 223, 2521, 4201, 21563, 37691, 737161, 1328521, 31463413, 57821173, 21404465, 39854897, 1267947073, 2383173185, 85428430547, 808549483039, 1535039635999, 2921975382559, 128230606647497, 245195521274057, 2348840786785261, 4508193056814061
Offset: 0

Views

Author

Vladimir Reshetnikov, Apr 08 2022

Keywords

Examples

			a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the denominators: 1, 1, 3, 13, 19, ...

Links

Crossrefs

Cf. A353250 (numerators).
Cf. A003149, A136128, A241519, A242376.

Programs

  • Mathematica
    Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 26}] // Denominator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 26}] // Denominator

Formula

a(n) = denominator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.
Showing 1-3 of 3 results.

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