A294521 -id:A294521 - cp's OEIS frontend

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

From Wolfdieter Lang, Nov 09 2017: (Start)
In the Downey et al. link this is the instance k = 5 of the formula given there for S_{2*k+2}. A simpler formula is given in the Koecher reference as (5/4)*v_5(1) on p. 192. See the Kotesovec formula given below.
The partial sums are given in A294520/A294521. (End)

			1.1779560579226638587351739680918874184458572345666798028425228573...

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

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.
Wikipedia, Polygonal number

Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.

RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 1/(5n^2 - 4n).
Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - Vaclav Kotesovec, Jul 04 2014
This is the value given in  the Koecher reference (see a comment above), and rewritten with the golden section phi = (1 + sqrt(5))/2 this becomes
  ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - Wolfdieter Lang, Nov 09 2017

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.

cp's OEIS Frontend

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

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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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Magma

Mathematica

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Formula

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.