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A241748 a(n) = n^2 + 12.

%I A241748 #35 Nov 29 2024 18:29:28
%S A241748 12,13,16,21,28,37,48,61,76,93,112,133,156,181,208,237,268,301,336,
%T A241748 373,412,453,496,541,588,637,688,741,796,853,912,973,1036,1101,1168,
%U A241748 1237,1308,1381,1456,1533,1612,1693,1776,1861,1948,2037,2128,2221,2316,2413,2512
%N A241748 a(n) = n^2 + 12.
%C A241748 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 = 3*r/4, r2 = r - r1 = r/4. See a comment on A114949 also for the MathWorld Pappus chain link. - _Wolfdieter Lang_, Jun 29 2015
%H A241748 Vincenzo Librandi, <a href="/A241748/b241748.txt">Table of n, a(n) for n = 0..1000</a>
%H A241748 Kival Ngaokrajang, <a href="/A241748/a241748.pdf">Illustration of Pappus chain (3/4, 1/4)</a>.
%H A241748 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A241748 G.f.: (12-23*x+13*x^2)/(1-x)^3.
%F A241748 a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
%F A241748 (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
%F A241748 From _Amiram Eldar_, Nov 02 2020: (Start)
%F A241748 Sum_{n>=0} 1/a(n) = (1 + sqrt(12)*Pi*coth(sqrt(12)*Pi))/24.
%F A241748 Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(12)*Pi*cosech(sqrt(12)*Pi))/24. (End)
%F A241748 E.g.f.: exp(x)*(12 + x + x^2). - _Elmo R. Oliveira_, Nov 29 2024
%t A241748 Table[n^2 + 12, {n, 0, 60}]
%o A241748 (Magma) [n^2+12: n in [0..60]];
%o A241748 (PARI) a(n)=n^2+12 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A241748 Cf. similar sequences listed in A114962.
%Y A241748 Cf. A114964 (see comment), A114949.
%K A241748 nonn,easy
%O A241748 0,1
%A A241748 _Vincenzo Librandi_, Apr 30 2014