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 1-2 of 2 results.

A275792 Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.

Original entry on oeis.org

1, 1, 5, 0, 9, 8, 2, 3, 6, 8, 0, 9, 4, 6, 7, 6, 3, 8, 6, 3, 6, 3, 6, 8, 9, 8, 9, 6, 9, 5, 2, 6, 7, 5, 0, 5, 8, 3, 0, 9, 6, 6, 7, 0, 9, 5, 5, 1, 8, 7, 4, 9, 1, 0, 9, 8, 3, 9, 6, 4, 5, 7, 8, 4, 5, 0, 5, 0, 4, 2, 6, 9, 1, 0, 9, 1, 3, 6, 6, 7, 4, 1, 4, 0, 9, 6, 6, 7, 5, 5, 3, 7, 0, 6, 3, 0, 5, 1, 5
Offset: 1

Views

Author

Wolfdieter Lang, Sep 12 2016

Keywords

Comments

See Table 1 of the Downey et al. link.
From Wolfdieter Lang, Nov 09 2017: (Start)
The general formula for S_{2*(k+1)} = Sum_{n>=0} 1/((n+1)*(k*n+1)) given in the Downey et al. link is a special case of the simpler formula for V(m,r) = Sum_{n>=0} 1/((n+1)*(m*n + r)), r = 1,2, ... ,m -1. V(m,r) = (m/(m-r))*v_m(r) in Koecher's notation. For this formula for m*v_m(r) see a comment in A294512.
The special case is m = k and r = 1, leading to S_{2*(k+1)} = V(k,1) = (log(k) + (Pi/2)*cot(Pi/k) - Sum_{j=1..k-1} cos(2*Pi*j/k)*log(2*sin(Pi*j/k)))/(k-1), for k >= 2.
S_14, for k=6, is then given by the formula below (also obtained from the more complicated formula of Downey et al.).
The partial sums are given in A294834/A294835.
(End)

Examples

			1.150982368094676386363689896952675058309...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.

Links

Crossrefs

Cf. A013661, A016627, A051866, A244641, A244645, A294512, A294834, A294835.

Programs

  • Magma
    SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024
  • Mathematica
    RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)
  • PARI
    2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017
  • SageMath
    numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024

Formula

Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.

A294835 Denominators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1), for k >= 0.

Original entry on oeis.org

1, 14, 546, 20748, 2593500, 26799500, 991581500, 85276009000, 5372388567000, 59096274237000, 3604872728457000, 241526472806619000, 17631432514883187000, 1392883168675771773000, 23679013867488120141000, 47358027734976240282000, 4593728690292695307354000, 157718018366715872219154000
Offset: 0

Views

Author

Wolfdieter Lang, Nov 20 2017

Keywords

Comments

The corresponding numerators are given in A294834. Details are found there.

Examples

			See A294834 for the rationals.

Crossrefs

Cf. A051866, A294834.

Programs

  • PARI
    a(n) = denominator(sum(k=0, n, 1/((k + 1)*(6*k + 1)))); \\ Michel Marcus, Nov 21 2017

Formula

a(n) = denominator(V(6,1;n)) with V(6,1;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 1)) = Sum_{k=0..n} 1/A051866(k+1) = (1/5)*Sum_{k=0..n} (1/(k + 1/6) - 1/(k + 1)). For the formula in terms of the digamma function see A294834.
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.