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A294834 Numerators of the partial sums of the reciprocals of the positive tetradecagonal numbers (k + 1)*(6*k + 1) = A051866(k+1). 3
1, 15, 599, 23035, 2900123, 30112021, 1117973277, 96393597197, 6084978910411, 67042215785861, 4094947551504521, 274661892011507657, 20068897076286721961, 1586702257063428405419, 26992510145660626515763, 54017546409271099350401, 5242487768036648180534897, 180077149085745155963315797 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
LINKS
Eric Weisstein's World of Mathematics, Digamma Function
FORMULA
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.
EXAMPLE
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.
MATHEMATICA
Table[Numerator[Sum[1/((k + 1)*(6*k + 1)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)
PROG
(PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(6*k + 1)))); \\ Michel Marcus, Nov 21 2017
(Magma) [Numerator((&+[1/((k + 1)*(6*k + 1)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018
CROSSREFS
Sequence in context: A179895 A343074 A232291 * A276489 A243227 A215899
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 20 2017
STATUS
approved

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Last modified April 16 05:35 EDT 2024. Contains 371697 sequences. (Running on oeis4.)