OFFSET
0,1
COMMENTS
3/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the counterclockwise Pappus chain of the arbelos with semicircle radii r, r1 = 3r/4, r2 = r - r1 = r/4. See a comment on A114949 also for the MathWorld Pappus chain link. - Wolfdieter Lang, Jun 29 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of Pappus chain (3/4, 1/4)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (12-23*x+13*x^2)/(1-x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
(2*n)*a(n) = (n+2)^3 + (n-2)^3; also, 2*a(n) = (n+sqrt(12))^2 + (n-sqrt(12))^2. - Bruno Berselli, Mar 13 2015
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(12)*Pi*coth(sqrt(12)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(12)*Pi*cosech(sqrt(12)*Pi))/24. (End)
MATHEMATICA
Table[n^2 + 12, {n, 0, 60}]
PROG
(Magma) [n^2+12: n in [0..60]];
(PARI) a(n)=n^2+12 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Apr 30 2014
STATUS
approved