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.

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)*(5*k + 1) = A051624(k+1), for k >= 0.

Original entry on oeis.org

1, 13, 49, 795, 84179, 366829, 11417459, 103067441, 4235695001, 97604192047, 1661825059679, 1663957022369, 101611584435869, 101706166053389, 7226964017429851, 17176158550059533, 154681745346189277, 6654999228519884521, 6658297729691103841, 21316057915886595965, 2153790894613123442641
Offset: 0

Views

Author

Wolfdieter Lang, Nov 15 2017

Keywords

Comments

The corresponding denominators are given in A294521.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,1].
The limit of the series is V(5,1) = lim_{n -> oo} V(5,1;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8, with the golden section phi:= (1 + sqrt(5))/2. The value is 1.17795605792266... given in A244649.

Examples

			The rationals V(5,1;n), n >= 0, begin: 1, 13/12, 49/44, 795/704, 84179/73920, 366829/320320, 11417459/9929920, 103067441/89369280, 4235695001/3664140480, 97604192047/84275231040, 1661825059679/1432678927680, ...
V(5,1;10^6) = 1.177956058 (Maple, 10 digits) to be compared with 1.177956058 obtained from V(5,1) given in A244649.

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.

Links

Crossrefs

Cf. A001620, A051624, A244649, A294512, A294516/A294517, A294521.

Programs

  • Magma
    [Numerator((&+[1/((k+1)*(5*k+1)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018
  • Mathematica
    Table[Numerator[Sum[1/((k + 1)*(5*k + 1)), {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Aug 29 2018 *)
  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ Michel Marcus, Nov 15 2017

Formula

a(n) = numerator(V(5,1;n)) with V(5,1;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 1)) = Sum_{k=0..n} 1/A051624(k+1) = (1/4)*Sum_{k=0..n} (1/(k + 1/5) - 1/(k+1)) = (-Psi(1/5) + Psi(n+6/5) - (gamma + Psi(n+2)))/4, with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294516 Numerators of the partial sums of the reciprocals of (k+1)*(4*k+3) = A033991(k+1), for k >= 0.

Original entry on oeis.org

1, 17, 67, 2087, 40577, 315967, 8627249, 539432053, 543008461, 7096662277, 306487877071, 14457409539227, 246534893826499, 49437672710843, 14617658229054773, 29294219493288391, 1966205309547985477, 139821581165897995307, 700098935135639210887, 55378426713778630607653, 4601722042202662057443599, 12144567347216934480292961
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2017

Keywords

Comments

The corresponding numerators are given in A294517.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [4,3].
The limit of the series is V(4,3) = lim_{n -> oo} V(4,3;n) = 3*log(2) - Pi/2 = 0.50864521488493930902... given in A294518.

Examples

			The rationals V(4,3;n), n >= 0, begin: 1/3, 17/42, 67/154, 2087/4620, 40577/87780, 315967/672980, 8627249/18170460, 539432053/1126568520, 543008461/1126568520, 7096662277/14645390760, 306487877071/629751802680, ...
V(4,3;10^4) = 0.508620219 (Maple, 10 digits) to be compared with 0.508645215 from V(4,3) given in A294518.

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Links

Crossrefs

Cf. A294512, A250551(n+1)/A294515(n) (V(4,1;n)), A294517, A294518.

Programs

  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(4*k + 3)))); \\ Michel Marcus, Nov 15 2017

Formula

a(n) = numerator(V(4,3;n)) with V(4,3;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 3)) = Sum_{k=0..n} 1/A033991(k+1) = Sum_{k=0..n} (4/(4*k + 3) - 1/(k+1)).
V(4,3;n) = 3*log(2) - Pi/2 + Psi(n+7/4) - Psi(n+2) with the digamma function Psi. Note that Psi(1) - Psi(3/4) = 3*log(2) - Pi/2. - Wolfdieter Lang, Nov 15 2017

A294518 Decimal expansion of 3*log(2) - Pi/2.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 07 2017

Keywords

Comments

This is the value of the series V(4,3) = lim_{n->oo} V(4,3;n) with the partial sums V(4,3;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 3)) = Sum_{k=0..n} 1/A033991(k+1) = Sum_{k=0..n} (4/(4*k + 3) - 1/(k+1)) = A294516(n)/A294517(n).
In the Koecher reference v_4(3) = (1/4)*V(4,3) = (3/4)*log(2) + Pi/8 = 0.1271613037212348272550...

Examples

			0.5086452148849393090203746727347782621279157033...

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189-193.

Crossrefs

Cf. A033991, A294516/A294517.

Programs

  • Mathematica
    RealDigits[3*Log[2] - Pi/2, 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

Formula

V(4,3) = 3*log(2) - Pi/2.
Equals Sum_{k>=2} zeta(k)/4^(k-1). - Amiram Eldar, May 31 2021
Showing 1-3 of 3 results.

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