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.

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A244639 Decimal expansion of the sum of the reciprocals of the heptagonal numbers (A000566).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jul 03 2014

Keywords

Comments

For the partial sums of one half of this series, that is Sum_{k>=0} 1/((k+1)*(5*k+2)), with value 0.6613896265611944283..., see A294826(n)/A294827(n), for n >= 0. - Wolfdieter Lang, Nov 16 2017

Examples

			1.32277925312238885674944226131008401652280117371392437228545762688516221076....

References

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

Links

Programs

  • Mathematica
    RealDigits[ Pi*Sqrt[25 - 10 Sqrt[5]]/15 + 2Log[5]/3 + (1 + Sqrt[5]) Log[ Sqrt[ 10 - 2 Sqrt[5]]/2]/3 + (1 - Sqrt[5]) Log[ Sqrt[ 10 + 2 Sqrt[5]]/2]/3, 10, 111][[1]] (* or *)
RealDigits[ Sum[2/(5 n^2 - 3 n), {n, 1, Infinity}], 10, 111][[1]]
  • PARI
    sumnumrat(2/n/(5*n-3),1) \\ Charles R Greathouse IV, Feb 08 2023

Formula

Equals Sum_{n>=1} 2/(5n^2 - 3n).
((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/3, with the golden section phi := (1 + sqrt(5))/2. This is (5/10)*v_5(2) given from the Koecher reference on p. 192 as ((5/2)*log(5) - sqrt(5)*log((1+sqrt(5))/2) + (1/5)*Pi*sqrt(5*(5-2*sqrt(5))))/3. Compare this with the number given in the Mathematica program. - Wolfdieter Lang, Nov 16 2017

A294826 Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)*(5*k + 2) = A135706(k+1) = 2*A000566(k+1), for k >= 0.

Original entry on oeis.org

1, 4, 151, 1315, 36698, 667109, 10749479, 399851303, 401511863, 18933826729, 246810236317, 4700047812703, 145981746528913, 9796912235587651, 9810925971351679, 9823210739716249, 403196782523223569, 11704197956499986461, 269433333504358946963, 5231145593209503407215, 747842028258712790473
Offset: 0

Views

Author

Wolfdieter Lang, Nov 16 2017

Keywords

Comments

The corresponding denominators are given in A294827.
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] = [5,2].
The limit of the series is V(5,2) = lim_{n -> oo} V(5,2;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/6, with the golden section phi:= (1 + sqrt(5))/2. The value is 0.661389626561... given by (1/2)*A244639.
In the Koecher reference v_5(2) = (3/5)*V(5,2) = 0.39683377593671665701 ...is given as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) + (Pi/10)*sqrt((5 - 2*sqrt(5))/5).

Examples

			The rationals V(5,2;n), n >= 0, begin: 1/2, 4/7, 151/252, 1315/2142, 36698/58905, 667109/1060290, 10749479/16964640, 399851303/627691680, 401511863/627691680, 18933826729/29501508960, 246810236317/383519616480, ...
V(5,2;10^6) = 0.6613894266 (Maple, 10 digits) to be compared with 0.6613896266 giving the 10 digit value of V(5,2) from (1/2)*A244649.

References

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

Links

Crossrefs

Cf. A001620, A000566, A135706, A294512, A294520/A294521 (V(5,1;n)), A244639.

Programs

  • Magma
    [Numerator((&+[1/((k+1)*(5*k+2)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018
  • Mathematica
    Table[Numerator[Sum[1/((k+1)*(5*k+2)), {k,0,n}]], {n,0,25}] (* G. C. Greubel, Aug 29 2018 *)
Accumulate[1/(2*PolygonalNumber[7,Range[30]])]//Numerator (* Harvey P. Dale, Aug 31 2023 *)
  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 2)))); \\ Michel Marcus, Nov 17 2017

Formula

a(n) = numerator(V(5,2;n)) with V(5,2;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 2)) = Sum_{k=0..n} 1/A135706(k+1) = (1/3)*Sum_{k=0..n} (1/(k + 2/5) - 1/(k+1)) = (-Psi(2/5) + Psi(n+7/5) - (gamma + Psi(n+2)))/3 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294828 Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 3) = A147874(k+2), for k >= 0.

Original entry on oeis.org

1, 19, 263, 815, 95597, 678149, 7531399, 18016577, 259695727, 4173941423, 222039686299, 2153029760377, 19428099753313, 331021112488901, 24211723390477517, 12126560607807901, 1008024074147249303, 168229609596886043, 1740462375346732831, 1219642439745618215
Offset: 0

Views

Author

Wolfdieter Lang, Nov 16 2017

Keywords

Comments

The corresponding denominators are given in A294829.
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] = [5,3].
The limit of the series is V(5,3) = lim_{n -> oo} V(5,3;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.48170177449... given in A294830.
In the Koecher reference v_5(3) = (2/5)*V(5,3) = 0.19268070979833151082... given there by ((1/4)*log(5) - (1/(2*sqrt(5)))*log((1+sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5)).

Examples

			The rationals V(5,3;n), n >= 0, begin: 1/3, 19/48, 263/624, 815/1872, 95597/215280, 678149/1506960, 7531399/16576560, 18016577/39369330, 259695727/564293730, 4173941423/9028699680, 222039686299/478521083040, 2153029760377/4625703802720, 19428099753313/41631334224480, ...
V(5,3;10^6) = 0.4817015746 to be compared with 0.4817017745 from A294830 with 10 digits.

References

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

Links

Crossrefs

Cf. A001620, A001622, A147874, A294512, A294826/A294827 (V(5,2;n)), A294829, A294830,

Programs

  • Maple
    map(numer,ListTools:-PartialSums([seq(1/(k+1)/(5*k+3),k=0..50)])); # Robert Israel, Nov 17 2017
  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ Michel Marcus, Nov 17 2017

Formula

a(n) = numerator(V(5,3;n)) with V(5,3;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 3)) = Sum_{k=0..n} 1/A147874(k+2) = (1/2)*Sum_{k=0..n} (1/(k + 3/5) - 1/(k+1)) = (-Psi(3/5) + Psi(n+8/5) - (gamma + Psi(n+2)))/2 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.
Showing 1-3 of 3 results.

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