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A158490 a(n) = 100*n^2 - 10.

%I A158490 #26 Jan 25 2025 14:47:18
%S A158490 90,390,890,1590,2490,3590,4890,6390,8090,9990,12090,14390,16890,
%T A158490 19590,22490,25590,28890,32390,36090,39990,44090,48390,52890,57590,
%U A158490 62490,67590,72890,78390,84090,89990,96090,102390,108890,115590,122490,129590,136890,144390,152090
%N A158490 a(n) = 100*n^2 - 10.
%C A158490 The identity (20*n^2 - 1)^2 - (100*n^2 - 10)*(2*n)^2 = 1 can be written as A158491(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158490 Vincenzo Librandi, <a href="/A158490/b158490.txt">Table of n, a(n) for n = 1..10000</a>
%H A158490 Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H A158490 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158490 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158490 G.f: 10*x*(-9-12*x+x^2)/(x-1)^3.
%F A158490 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158490 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(10))*Pi/sqrt(10))/20.
%F A158490 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(10))*Pi/sqrt(10) - 1)/20. (End)
%F A158490 From _Elmo R. Oliveira_, Jan 17 2025: (Start)
%F A158490 E.g.f.: 10*(exp(x)*(10*x^2 + 10*x - 1) + 1).
%F A158490 a(n) = 10*A158447(n). (End)
%t A158490 LinearRecurrence[{3,-3,1},{90,390,890},20]
%t A158490 100*Range[40]^2-10 (* _Harvey P. Dale_, Apr 03 2019 *)
%o A158490 (Magma) I:=[90, 390, 890]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
%o A158490 (PARI) a(n)=100*n^2-10 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A158490 Cf. A005843, A158447, A158491.
%K A158490 nonn,easy
%O A158490 1,1
%A A158490 _Vincenzo Librandi_, Mar 20 2009