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A158547 a(n) = 24*n^2 + 1.

%I A158547 #37 Feb 04 2025 05:40:05
%S A158547 1,25,97,217,385,601,865,1177,1537,1945,2401,2905,3457,4057,4705,5401,
%T A158547 6145,6937,7777,8665,9601,10585,11617,12697,13825,15001,16225,17497,
%U A158547 18817,20185,21601,23065,24577,26137,27745,29401,31105,32857,34657,36505,38401,40345
%N A158547 a(n) = 24*n^2 + 1.
%C A158547 The identity (24*n^2 + 1)^2 - (144*n^2 + 12)*(2*n)^2 = 1 can be written as a(n)^2 - A158546(n)*A005843(n)^2 = 1.
%H A158547 Vincenzo Librandi, <a href="/A158547/b158547.txt">Table of n, a(n) for n = 0..10000</a>
%H A158547 Leo Tavares, <a href="/A158547/a158547.jpg">Illustration: Stellar Stars</a>.
%H A158547 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158547 G.f.: (1 + 22*x + 25*x^2)/(1-x)^3.
%F A158547 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158547 From _Amiram Eldar_, Mar 02 2023: (Start)
%F A158547 Sum_{n>=0} 1/a(n) = 1/2 + coth(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)).
%F A158547 Sum_{n>=0} (-1)^n/a(n) = 1/2 + cosech(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)). (End)
%F A158547 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158547 E.g.f.: exp(x)*(1 + 24*x + 24*x^2).
%F A158547 a(n) = A227776(2*n). (End)
%t A158547 LinearRecurrence[{3, -3, 1}, {1, 25, 97}, 50] (* _Vincenzo Librandi_, Feb 14 2012 *)
%o A158547 (Magma) I:=[1, 25, 97]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 14 2012
%o A158547 (PARI) for(n=0, 40, print1(24*n^2+1", ")); \\ _Vincenzo Librandi_, Feb 14 2012
%Y A158547 Cf. A005843, A008594, A158546, A227776.
%K A158547 nonn,easy
%O A158547 0,2
%A A158547 _Vincenzo Librandi_, Mar 21 2009
%E A158547 Comment rewritten, a(0) added by _R. J. Mathar_, Oct 16 2009