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A144555 a(n) = 14*n^2.

%I A144555 #28 Nov 30 2024 08:50:09
%S A144555 0,14,56,126,224,350,504,686,896,1134,1400,1694,2016,2366,2744,3150,
%T A144555 3584,4046,4536,5054,5600,6174,6776,7406,8064,8750,9464,10206,10976,
%U A144555 11774,12600,13454,14336,15246,16184,17150,18144,19166,20216,21294,22400,23534,24696
%N A144555 a(n) = 14*n^2.
%C A144555 Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - _Omar E. Pol_, Sep 10 2011
%H A144555 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A144555 a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - _Omar E. Pol_, Jan 01 2009
%F A144555 a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - _Vincenzo Librandi_, Nov 25 2010
%F A144555 From _Amiram Eldar_, Feb 03 2021: (Start)
%F A144555 Sum_{n>=1} 1/a(n) = Pi^2/84.
%F A144555 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
%F A144555 Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
%F A144555 Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
%F A144555 From _Elmo R. Oliveira_, Nov 30 2024: (Start)
%F A144555 G.f.: 14*x*(1 + x)/(1-x)^3.
%F A144555 E.g.f.: 14*x*(1 + x)*exp(x).
%F A144555 a(n) = n*A008596(n) = A195145(2*n).
%F A144555 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A144555 Table[14*n^2, {n, 0, 45}] (* _Amiram Eldar_, Feb 03 2021 *)
%o A144555 (PARI) A144555(n)=14*n^2 \\ _M. F. Hasler_, Oct 31 2014
%Y A144555 See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
%Y A144555 Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
%Y A144555 Cf. A008596, A195145.
%K A144555 nonn,easy
%O A144555 0,2
%A A144555 _N. J. A. Sloane_, Jan 01 2009