A222465 a(n) = 4*n^2 + 3.

3, 7, 19, 39, 67, 103, 147, 199, 259, 327, 403, 487, 579, 679, 787, 903, 1027, 1159, 1299, 1447, 1603, 1767, 1939, 2119, 2307, 2503, 2707, 2919, 3139, 3367, 3603, 3847, 4099, 4359, 4627, 4903, 5187, 5479, 5779, 6087, 6403, 6727, 7059, 7399, 7747, 8103, 8467, 8839

Offset: 0

Author

Wolfdieter Lang, Mar 01 2013

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

Cf. A016742, A114949.

Programs

Maple

A222465(n):=n->4*n^2 + 3; seq(A222465(n), n=0..50); # Wesley Ivan Hurt, Feb 06 2014

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 *)

PARI

a(n)=4*n^2+3 \\ Charles R Greathouse IV, Aug 20 2013

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)

E.g.f.: exp(x)*(3 + 4*x + 4*x^2). - Elmo R. Oliveira, Jan 17 2025