|
| |
|
|
A275792
|
|
Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.
|
|
3
|
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
See Table 1 of the Downey et al. link.
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.).
(End)
|
|
|
REFERENCES
|
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.
|
|
|
EXAMPLE
|
1.150982368094676386363689896952675058309...
|
|
|
MATHEMATICA
|
RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)
|
|
|
PROG
|
(PARI) 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017
(Magma) SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024
(SageMath) numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
