OFFSET
0,1
COMMENTS
2/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the clockwise Pappus chain of the arbelos with semicircle radii r, r1 = 2r/3, r2 = r/3. See the MathWorld link for Pappus chain (there only the counterclockwise chain is shown). The counterclockwise chain companion has circle radii R(n)/r = 2/A114949(n), n>=0.
Binomial transform of (3, 4, 8, 0, 0, 0, 0, 0, 0, 0, ...). - Philippe Deléham, Mar 07 2013
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of clockwise Pappus chain
Eric Weisstein's World of Mathematics, Pappus chain
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 4*n^2 + 3, n >= 0.
O.g.f.: (3 - 2*x + 7*x^2)/(1-x)^3.
a(n) = A016742(n) + 3. - Omar E. Pol, Mar 02 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0) = 3, a(1) = 7, a(2) = 19. - Philippe Deléham, Mar 05 2013
From Amiram Eldar, Jul 11 2020: (Start)
Sum_{n>=0} 1/a(n) = 1/6 + sqrt(3)*Pi*coth(sqrt(3)*Pi/2)/12.
Sum_{n>=0} (-1)^n/a(n) = 1/6 + sqrt(3)*Pi*cosech(sqrt(3)*Pi/2)/12. (End)
EXAMPLE
The dimensionless radii R(n)/r of the clockwise Pappus chain for the arbelos (r,r1,r2=r-r1) = r*(1,2/3,1/3) are [2/3, 2/7, 2/19, 2/39, 2/67, 2/103, 2/147, 2/199 ,...], for n >= 0. The circle for n=0 has radius r1=2/3 and center (2/3,0) with the origin at the left tip of the arbelos. The n=1 circle coincides with the one of the counterclockwise companion chain.
MAPLE
MATHEMATICA
Table[4 n^2 + 3, {n, 0, 50}] (* Wesley Ivan Hurt, Feb 06 2014 *)
Array[4 #^2 + 3 &, 44, 0] (* Luiz Roberto Meier, Jan 22 2015 *)
PROG
(PARI) a(n)=4*n^2+3 \\ Charles R Greathouse IV, Aug 20 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 01 2013
STATUS
approved