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%I #67 Feb 05 2024 02:25:12
%S 6,7,10,15,22,31,42,55,70,87,106,127,150,175,202,231,262,295,330,367,
%T 406,447,490,535,582,631,682,735,790,847,906,967,1030,1095,1162,1231,
%U 1302,1375,1450,1527,1606,1687,1770,1855,1942,2031,2122,2215,2310,2407,2506
%N a(n) = n^2 + 6.
%C 2/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 = 2r/3, r2 = r - r1 = r/3. See the MathWorld link for such a Pappus chain. The clockwise chain companion has circle radii R'(n)/r = 2/A222465(n), n>=0. - _Wolfdieter Lang_, Mar 01 2013
%H Ivan Panchenko, <a href="/A114949/b114949.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PappusChain.html">Pappus chain</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*a(n - 1) - 3*a(n - 2) + a(n - 3). - _R. J. Mathar_, May 17 2009
%F G.f.: -(6 - 11*x + 7*x^2)/(x - 1)^3. - _R. J. Mathar_, May 17 2009
%F a(n) = 2*n + a(n - 1) - 1, with n>0, a(0)=6. - _Vincenzo Librandi_, Nov 13 2010
%F a(n) = A000290(n) + 6. - _Omar E. Pol_, Mar 02 2013
%F (5n)*a(n) = (n-2)^3 + (n-1)^3 + n^3 + (n+1)^3 + (n+2)^3. - _Bruno Berselli_, May 12 2014
%F For n>=1, a(n) = (A016742(n) + A082044(n) - 1) / A000290(n). - _Bruce J. Nicholson_, Apr 19 2017
%F From _Amiram Eldar_, Nov 02 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/12.
%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/12. (End)
%F From _Amiram Eldar_, Feb 05 2024: (Start)
%F Product_{n>=0} (1 - 1/a(n)) = sqrt(5/6)*sinh(sqrt(5)*Pi)/sinh(sqrt(6)*Pi).
%F Product_{n>=0} (1 + 1/a(n)) = sqrt(7/6)*sinh(sqrt(7)*Pi)/sinh(sqrt(6)*Pi). (End)
%e The arbelos chain defined in a comment above has circle radii [1/3, 2/7, 1/5, 2/15, 1/11, 2/31, 1/21, 2/55, 1/35, 2/87, 1/53,...], for n >= 0. - _Wolfdieter Lang_, Mar 01 2013
%p A114949:=n->n^2+6: seq(A114949(n), n=0..100); # _Wesley Ivan Hurt_, Apr 28 2017
%t Range[0, 49]^2 + 6 (* _Alonso del Arte_, Jan 30 2013 *)
%Y Cf. A002522, A059100, A087475, A117619, A117950, A117951, A114964 (see comment), A222465, A016742, A082044, A000290.
%K nonn,easy
%O 0,1
%A _Cino Hilliard_, Feb 21 2006
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