A294512 -id:A294512 - cp's OEIS frontend

A294515 Denominators of partial sums of the reciprocals of the decagonal numbers.

1, 10, 270, 7020, 119340, 835380, 4176900, 242260200, 888287400, 32866633800, 1347531985800, 4042595957400, 28298171701800, 1499803100195400, 28496258903712600, 3476543586252937200, 3476543586252937200, 26653500827939185200, 1945705560439560519600, 1945705560439560519600, 52534050131868134029200

Offset: 0

Wolfdieter Lang, Nov 02 2017

The corresponding numerators are given by A250551(n+1), n >= 0.
The positive decagonal numbers are A001107(k+1) = (k + 1)*(4*k + 1), k >= 0.
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] = [4,1].
The limit of the series is V(4,1) = lim_{n -> oo} V(4,1;n) = log(2) + Pi/6 = 1.216745956158244182... given in A244647.

			The rationals V(4,1;n), n >= 0, begin: 1, 11/10, 307/270, 8117/7020, 139393/119340, 982381/835380, 4935773/4176900, 287319059/242260200, 1056494083/888287400, 39179109811/32866633800, ...
V(4,1;10^4) = 1.216720959 (Maple, 10 digits) to be compared with 1.216745956 from V(4,1) from A244647.

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

Robert Israel, Table of n, a(n) for n = 0..866

Cf. A001107, A244647, A250550, A250551.

map(denom,ListTools:-PartialSums([seq(1/((k+1)*(4*k+1)),k=0..50)])); # Robert Israel, Nov 08 2017

Denominator@ Accumulate@ Array[1/PolygonalNumber[10, #] &, 23] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = denominator(V(4,1;n)) with V(4,1;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 1)) = Sum_{k=0..n} 1/A001107(n+1) = (1/3)*Sum_{k=0..n} (4/(4*k + 1) - 1/(k+1)).
a(n) = A250550(n+1)/(n+1) [conjecture].
In the Koecher reference v_4(1) = (3/4)*V(4,1) = (3/4)*log(2) + Pi/8 = 0.91255946711868313687... .

A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)(5k + 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

Wolfdieter Lang, Nov 15 2017

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.

			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.

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

[Numerator((&+[1/((k+1)*(5*k+1)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

Table[Numerator[Sum[1/((k + 1)*(5*k + 1)), {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Aug 29 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ Michel Marcus, Nov 15 2017

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.

A294826 Numerators of the partial sums of the reciprocals of twice the heptagonal numbers (k + 1)(5k + 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

Wolfdieter Lang, Nov 16 2017

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).

			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.

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

[Numerator((&+[1/((k+1)*(5*k+2)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

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 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 2)))); \\ Michel Marcus, Nov 17 2017

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)(5k + 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

Wolfdieter Lang, Nov 16 2017

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)).

			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.

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

Robert Israel, Table of n, a(n) for n = 0..891
Eric Weisstein's World of Mathematics, Digamma Function

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

map(numer,ListTools:-PartialSums([seq(1/(k+1)/(5*k+3),k=0..50)])); # Robert Israel, Nov 17 2017

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ Michel Marcus, Nov 17 2017

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.

A250401 Denominator of the harmonic mean of the first n nonzero octagonal numbers.

Original entry on oeis.org

1, 9, 197, 503, 6623, 17813, 340527, 3763087, 169947523, 170436583, 5295982873, 90208585541, 3343268872217, 3348036962687, 144143598106421, 1659445372263179, 11627213232841853, 3879029288899801, 352907045903771, 10241306344308349, 208368821623076563

Offset: 1

Colin Barker, Nov 21 2014

a(n+1), for n >= 0, is also the numerator of the partial sums of the reciprocal octagonal numbers Sum_{k=0..n} 1/((k + 1)*(3*k + 1)) with the denominators given in A294512(n) [assuming that n+1 divides A250400(n+1) to give A294512(n) for n >= 0]. - Wolfdieter Lang, Nov 01 2017

			a(3) = 197 because the octagonal numbers A000567(n), for n = 1..3, are [1,8,21], and 3/(1/1 + 1/8 + 1/21) = 504/197.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000567 (octagonal numbers), A250400 (numerators), A294512.

f:= n -> denom(n/add(1/(k*(3*k-2)),k=1..n)):
map(f, [$1..40]); # Robert Israel, Nov 01 2017

With[{s = Array[PolygonalNumber[8, #] &, 21]}, Denominator@ Array[HarmonicMean@ Take[s, #] &, Length@ s]] (* Michael De Vlieger, Nov 01 2017 *)

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(n=1, #s, s[n]=denominator(harmonicmean(vector(n, k, 3*k^2-2*k)))); s

Denominator of 12*n/(Pi*sqrt(3) + 9*log(3) + 6*Psi(n+1/3) - 6*Psi(n+1)). - Robert Israel, Nov 01 2017

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

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.

A294516 Numerators of the partial sums of the reciprocals of (k+1)(4k+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

Wolfdieter Lang, Nov 07 2017

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.

			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.

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

Eric Weisstein's World of Mathematics, Digamma Function

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

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

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

A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

Original entry on oeis.org

1, 11, 83, 410, 16799, 495151, 8516747, 55850623, 309309419, 1088610631, 6561497681, 777210281963, 12475578306953, 287734917200239, 10671842976127147, 844855135994501953, 846430303832665873, 75457260356268267017, 3551759427031132995079, 711302288219532928235, 712163917143684270659

Offset: 0

Wolfdieter Lang, Nov 18 2017

The corresponding denominators are given in A294832.
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] = [5,4].
The limit of the series is V(5,4) = lim_{n -> oo} V(5,4;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.3877929018046... given in A294833.
In the Koecher reference v_5(4) = (1/5)*V(5,4) = 0.07755858036... given there by (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5).

			The rationals V(5,4;n), n >= 0, begin:1/4, 11/36, 83/252, 410/1197, 16799/47880, 495151/1388520, 8516747/23604840, 55850623/153431460, 309309419/843873030, 1088610631/2953555605, 6561497681/17721333630, 777210281963/2091117368340, 12475578306953/33457877893440, ...
V(5,4;10^6) = 0.3877927018 (Maple 10 digits) to be compared with 0.3877929018 obtained from A294833 with 10 digits.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

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

Cf. A001622, 2*A005476, A294512, A294828/A294829 (V(5,3;n)), A294832, A294833.

[Numerator((&+[1/((k + 1)*(5*k + 4)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018

Table[Numerator[Sum[1/((k + 1)*(5*k + 4)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ Michel Marcus, Nov 19 2017

a(n) = numerator(V(5,4;n)) with V(5,4;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 4)) = Sum_{k=0..n} 1/(2*A005476(k+1)) = Sum_{k=0..n} (1/(k + 4/5) - 1/(k+1)) = -Psi(4/5) + Psi(n+9/5) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

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.

A294964 Numerators of the partial sums of the reciprocals of the positive numbers (k + 1)(6k + 5) = A049452(k+1).

Original entry on oeis.org

1, 27, 1487, 71207, 423323, 5021921, 208393341, 19767960169, 9496615779853, 112702096556215, 7360072449683999, 524616965933727859, 526363371877036219, 43813027890740553917, 781806518388353706041, 148866078528885256002173, 15064339628673236669081953, 538212602352090865654383697

Offset: 0

Wolfdieter Lang, Nov 27 2017

The corresponding denominators are given in A294965.
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] = [6,5].
The limit of the series is V(6,5) = lim_{n -> oo} V(6,5;n) = . The value is (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = 0.3135137477... given in A294966.

			The rationals V(6,5;n), n >= 0, begin: 1/5, 27/110, 1487/5610, 71207/258060, 423323/1496748, 5021921/17462060, 208393341/715944460, 19767960169/67298779240, 9496615779853/32101517697480, ...
V(6,5;10^6) = 0.313513577 (Maple, 10 digits) to be compared with the rounded ten digits 0.3135137478 obtained from V(6,5) given in A294966.

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

Robert Israel, Table of n, a(n) for n = 0..640
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001620, A049452, A294512, A294966

[Numerator((&+[1/((k+1)*(6*k+5)): k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 29 2018

map(numer,  ListTools:-PartialSums([seq(1/(k+1)/(6*k+5),k=0..20)])); # Robert Israel, Nov 29 2017

Table[Numerator[Sum[1/((k+1)*(6*k+5)), {k,0,n}]], {n,0,25}] (* G. C. Greubel, Aug 29 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 5)))); \\ Michel Marcus, Nov 27 2017

a(n) = numerator(V(6,5;n)) with V(6,5;n) = Sum_{k=0..n} 1/((k + 1)*(6*k + 5)) = Sum_{k=0..n} 1/A049452(k+1) = Sum_{k=0..n} (1/(k + 5/6) - 1/(k + 1)) = -Psi(5/6) + Psi(n+11/6) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

cp's OEIS Frontend

A294515 Denominators of partial sums of the reciprocals of the decagonal numbers.

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

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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.

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

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A250401 Denominator of the harmonic mean of the first n nonzero octagonal numbers.

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

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

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

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

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

A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

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

A294964 Numerators of the partial sums of the reciprocals of the positive numbers (k + 1)(6k + 5) = A049452(k+1).