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 1-4 of 4 results.

A101028 Numerator of partial sums of a certain series. First member (m = 2) of a family.

Original entry on oeis.org

1, 11, 79, 479, 5297, 69071, 69203, 471181, 8960447, 44831407, 1031626241, 5160071143, 15484789693, 64166447971, 1989542332021, 3979714828967, 27861681000449, 1030996803010973, 1031094241305773, 42278288849598913, 1818093633186492859, 1818204269645957299, 85460151199040573933
Offset: 1

Views

Author

Wolfdieter Lang, Dec 17 2004

Keywords

Comments

The denominators are given in A101029.
The limit s = lim_{n -> infinity} s(n) with s(n) defined below equals 3*Sum_{k >= 1} zeta(2*k+1)/2^(2*k) with Euler's (or Riemann's) zeta function. This limit is 3*(2*log(2)-1) = 1.158883083...; see the Abramowitz-Stegun reference p. 259, eq. 6.3.15 with z = 1/2 together with p. 258, eqs. 6.3.5 and 6.3.3.
This is the first member (m = 2) of a family of rational partial sum sequences s(n,m) = (m-1)*m*(m+1)*Sum_{k = 1..n} 1/((m*k-1)*(m*k)*(m*k+1)) which have limit s(m) = lim_{n -> infinity} s(n,m) = -(gamma + Psi(1/m) + m/2 + Pi*cot(Pi*x)/2), with the Euler-Mascheroni constant gamma and the digamma function Psi. The same limit is reached by (m^2-1)*Sum_{k >= 0} zeta(2*k+1)/m^(2*k).
From Peter Bala, Feb 17 2022: (Start)
Let F(n) = (6*n/(2*n-1))*( 1/(1*2)*(n-1)/n - 1/(2*3)*(n-1)*(n-2)/(n*(n+1)) + 1/(3*4)*(n-1)*(n-2)*(n-3)/(n*(n+1)*(n+2)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/(n*(n+1)*(n+2)*(n+3)) + ...). Then F(n+1) = 6*Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)). Cf. A082687.
This identity allows us to extend the definition of Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)) to non-integral values of n. (End)

Examples

			s(3)= 6*(1/(1*2*3)+ 1/(3*4*5) + 1/(5*6*7)) = 79/70, hence a(3)=79 and A101029(3)=70.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 258-259.
  • Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Links

Crossrefs

Cf. A101627 (m=3), A101629 (m=4), A101631 (m=5).
Cf. A082687, A101029 (denominators).

Programs

  • PARI
    a(n) = numerator(6*sum(k=1, n, 1/((2*k-1)*(2*k)*(2*k+1)))); \\ Michel Marcus, Feb 28 2022

Formula

a(n) = numerator(s(n)) with s(n) = 6*Sum_{k = 1..n} 1/((2*k-1)*(2*k)*(2*k+1)).

Extensions

More terms from Michel Marcus, Feb 28 2022

A101627 Numerator of partial sums of a certain series.

Original entry on oeis.org

1, 39, 241, 34883, 14039, 1516871, 7601151, 875425657, 7887002813, 7095769757767, 14199583385459, 75087685321529, 75113436870869, 927229349730873529, 927436191807263569, 305182576081725442901, 23479178371879154033, 37713848011377144613, 37717984058802320713, 135759786815564675620247
Offset: 1

Views

Author

Wolfdieter Lang, Dec 23 2004

Keywords

Comments

The denominators are given in A101628.
Second member (m=3) of a family defined in A101028.
The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 8*sum(zeta(2*k+1)/3^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 12*(log(3)-1) = 1.18334746...; see the Abramowitz-Stegun (given in A101028) reference p. 259, eq. 6.3.15 with z=1/3 together with p. 258.

Examples

			s(3)= 24*(1/(2*3*4)+ 1/(5*6*7) + 1/(8*9*10)) = 241/210, hence a(3)=241 and A101628(3)=210.

Links

Crossrefs

Cf. A101028 (m=2), A101629 (m=4), A101631 (m=5).
Cf. A101628 (denominators).

Programs

  • Mathematica
    Numerator[Accumulate[Table[8/(9k^3-k),{k,20}]]]
  • PARI
    a(n) = numerator(24*sum(k=1, n, 1/((3*k-1)*(3*k)*(3*k+1))));

Formula

a(n)=numerator(s(n)) with s(n)=24*sum(1/((3*k-1)*(3*k)*(3*k+1)), k=1..n).

Extensions

More terms from Michel Marcus, Mar 01 2022

A101631 Numerator of partial sums of a certain series.

Original entry on oeis.org

1, 37, 1069, 20575, 1346153, 1214756107, 20699705479, 850029466379, 19572345658457, 137116980686111, 411600123273343, 1482039573988769177, 456179332236626381, 32398234503565880731, 1199020509231104363863
Offset: 1

Views

Author

Wolfdieter Lang, Dec 23 2004

Keywords

Comments

The denominators are given in A101632.
Third member (m=5) of a family defined in A101028.
The limit s = lim_{n->oo} s(n) with the s(n) defined below equals 24*Sum_{k>=1} zeta(2*k+1)/5^(2*k) with Euler's (or Riemann's) zeta function. This limit is -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) = 1.1954056019...; see a comment in A101028 following from the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/5 together with p. 258.

Examples

			s(3) = 120*(1/(4*5*6) + 1/(9*10*11) + 1/(14*15*16)) = 1069/924, hence a(3)=1069 and A101632(3)=924.

Links

Crossrefs

Cf. A101028, A101627, A101629, members 2, 3, 4, resp.

Formula

a(n) = numerator(s(n)) where s(n) = 120*Sum_{k=1..n} 1/((5*k-1)*(5*k)*(5*k+1)) = 24*Sum_{k=1..n} 1/((5*k-1)*k*(5*k+1)).

A101630 Denominator of partial sums of a certain series.

Original entry on oeis.org

1, 42, 6006, 204204, 3879876, 446185740, 38818159380, 2406725881560, 89048857617720, 3651003162326520, 156993135980040360, 51650741737433278440, 2737489312083963757320, 2737489312083963757320
Offset: 1

Views

Author

Wolfdieter Lang, Dec 23 2004

Keywords

Comments

The numerators are given in A101629.
a(13) and a(14) are indeed identical, namely 2737489312083963757320.

Formula

a(n)=denominator(s(n)) with s(n)=15*sum(1/((4*k-1)*k*(4*k+1)), k=1..n).
Showing 1-4 of 4 results.

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