This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A222465 #52 Feb 16 2025 08:33:19
%S A222465 3,7,19,39,67,103,147,199,259,327,403,487,579,679,787,903,1027,1159,
%T A222465 1299,1447,1603,1767,1939,2119,2307,2503,2707,2919,3139,3367,3603,
%U A222465 3847,4099,4359,4627,4903,5187,5479,5779,6087,6403,6727,7059,7399,7747,8103,8467,8839
%N A222465 a(n) = 4*n^2 + 3.
%C A222465 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.
%C A222465 Binomial transform of (3, 4, 8, 0, 0, 0, 0, 0, 0, 0, ...). - _Philippe Deléham_, Mar 07 2013
%H A222465 Ivan Panchenko, <a href="/A222465/b222465.txt">Table of n, a(n) for n = 0..1000</a>
%H A222465 Kival Ngaokrajang, <a href="/A222465/a222465.pdf">Illustration of clockwise Pappus chain</a>.
%H A222465 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PappusChain.html">Pappus chain</a>.
%H A222465 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A222465 a(n) = 4*n^2 + 3, n >= 0.
%F A222465 O.g.f.: (3 - 2*x + 7*x^2)/(1-x)^3.
%F A222465 a(n) = A016742(n) + 3. - _Omar E. Pol_, Mar 02 2013
%F A222465 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
%F A222465 From _Amiram Eldar_, Jul 11 2020: (Start)
%F A222465 Sum_{n>=0} 1/a(n) = 1/6 + sqrt(3)*Pi*coth(sqrt(3)*Pi/2)/12.
%F A222465 Sum_{n>=0} (-1)^n/a(n) = 1/6 + sqrt(3)*Pi*cosech(sqrt(3)*Pi/2)/12. (End)
%F A222465 E.g.f.: exp(x)*(3 + 4*x + 4*x^2). - _Elmo R. Oliveira_, Jan 17 2025
%e A222465 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.
%p A222465 A222465(n):=n->4*n^2 + 3; seq(A222465(n), n=0..50); # _Wesley Ivan Hurt_, Feb 06 2014
%t A222465 Table[4 n^2 + 3, {n, 0, 50}] (* _Wesley Ivan Hurt_, Feb 06 2014 *)
%t A222465 Array[4 #^2 + 3 &, 44, 0] (* _Luiz Roberto Meier_, Jan 22 2015 *)
%o A222465 (PARI) a(n)=4*n^2+3 \\ _Charles R Greathouse IV_, Aug 20 2013
%Y A222465 Cf. A016742, A114949.
%K A222465 nonn,easy
%O A222465 0,1
%A A222465 _Wolfdieter Lang_, Mar 01 2013