A365523 - OEIS

%I #15 Nov 21 2024 09:27:10

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

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

%U 1,7,3,1,8,1,8,1,6,8,2,9,3,6,3,5,1,7,9,9,6,1,9,7,8,5,1,2,1,2,5,2,5,2,0,0,8,8,8,6,1,2

%N Decimal expansion of 6*log(2) - 4.

%C This sequence is also the decimal expansion of Sum_{k>=1} (-1)^(k+1)*f(k), where f(k) = (4*k^2 - 2*k)/(k^2 + k) is the ratio between the k-th hexagonal and triangular numbers.

%H Michael Ian Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">A catalog of the real numbers</a> (2011), p. 219.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Sum_{k>=1} k/(2^k*(k + 1)*(k + 2)) [Shamos].

%F Equals Sum_{k>=1} (-1)^(k+1)*(4*k^2 - 2*k)/(k^2 + k).

%e 0.15888308335967185650339272874905940845300080616153...

%t RealDigits[6*Log[2] - 4, 10 , 100][[1]] (* _Amiram Eldar_, Sep 08 2023 *)

%o (PARI) 6*log(2)-4

%Y Cf. A002162. Essentially the same as A016687.

%Y Cf. A000217, A000384.

%K nonn,cons

%O 0,2

%A _Claude H. R. Dequatre_, Sep 08 2023