A294835 -id:A294835 - cp's OEIS frontend

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

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

Wolfdieter Lang, Sep 12 2016

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)

			1.150982368094676386363689896952675058309...

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.

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

SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024

RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017

numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024

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

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1).

Original entry on oeis.org

1, 15, 599, 23035, 2900123, 30112021, 1117973277, 96393597197, 6084978910411, 67042215785861, 4094947551504521, 274661892011507657, 20068897076286721961, 1586702257063428405419, 26992510145660626515763, 54017546409271099350401, 5242487768036648180534897, 180077149085745155963315797

Offset: 0

Wolfdieter Lang, Nov 20 2017

The corresponding denominators are given in A294835.
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] = [6,1].
The limit of the series is V(6,1) = lim_{n -> oo} V(6,1;n) = (3/10)*log(3) + (2/5)*log(2) + (1/10)*Pi*sqrt(3). The value is 1.150982368094676386... given in A275792.

			The rationals V(6,1;n), n >= 0, begin: 1, 15/14, 599/546, 23035/20748, 2900123/2593500, 30112021/26799500, 1117973277/991581500, 96393597197/85276009000, 6084978910411/5372388567000, 67042215785861/59096274237000, 4094947551504521/3604872728457000, ...
V(6,1;10^6) = 1.150982200 (Maple, 10 digits) to be compared with the ten digits 1.150982368 obtained from V(6,1) given in A275792.

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

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A051866, A275792, A294512, A294835.

[Numerator((&+[1/((k + 1)*(6*k + 1)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018

Table[Numerator[Sum[1/((k + 1)*(6*k + 1)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)

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

a(n) = numerator(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)) = (-Psi(1/6) + Psi(n+7/6) - (gamma + Psi(n+2)))/5 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

cp's OEIS Frontend

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

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A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)(6k + 1) = A051866(k+1).