A294827 -id:A294827

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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.

+0
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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).

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..600

Eric Weisstein's World of Mathematics, Digamma Function

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.

EXAMPLE

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.

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 *)

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 16 2017

STATUS

approved

A244639

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

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9

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..105.

L. Downey, B. W. Ong, and J. A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J. 39, no. 8, 2008, 391-394.

Society for Industrial and Applied Mathematics, Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss

Wikipedia, Heptagonal Number

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

EXAMPLE

1.32277925312238885674944226131008401652280117371392437228545762688516221076....

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]]

PROG

(PARI) sumnumrat(2/n/(5*n-3), 1) \\ Charles R Greathouse IV, Feb 08 2023

KEYWORD

nonn,cons,easy

AUTHOR

Robert G. Wilson v, Jul 03 2014

STATUS

approved

A294828

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

+0
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The corresponding denominators are given in A294829.

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)).

REFERENCES

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..891

Eric Weisstein's World of Mathematics, Digamma Function

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.

EXAMPLE

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.

MAPLE

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 16 2017

STATUS

approved

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