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A158447 a(n) = 10*n^2 - 1.

%I A158447 #35 Aug 30 2022 13:56:00
%S A158447 9,39,89,159,249,359,489,639,809,999,1209,1439,1689,1959,2249,2559,
%T A158447 2889,3239,3609,3999,4409,4839,5289,5759,6249,6759,7289,7839,8409,
%U A158447 8999,9609,10239,10889,11559,12249,12959,13689,14439,15209,15999,16809,17639
%N A158447 a(n) = 10*n^2 - 1.
%C A158447 Sequence found by reading the line from 9, in the direction 9, 39, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Jul 18 2012
%H A158447 Vincenzo Librandi, <a href="/A158447/b158447.txt">Table of n, a(n) for n = 1..10000</a>
%H A158447 Vincenzo Librandi, <a href="https://web.archive.org/web/20110710101009/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>.
%H A158447 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158447 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158447 G.f: x*(9 + 12*x - x^2)/(1 - x)^3.
%F A158447 a(n) = A033583(n) - 1. - _Omar E. Pol_, Jul 18 2012
%F A158447 From _Amiram Eldar_, Feb 04 2021: (Start)
%F A158447 Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(10))*cot(Pi/sqrt(10)))/2.
%F A158447 Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(10))*csc(Pi/sqrt(10)) - 1)/2.
%F A158447 Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(10))*csc(Pi/sqrt(10)).
%F A158447 Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(10))*sin(Pi/sqrt(5))/sqrt(2). (End)
%F A158447 E.g.f.: exp(x)*(10*x^2 + 10*x - 1) + 1. - _Stefano Spezia_, Aug 25 2022
%p A158447 A158447:=n->10*n^2-1: seq(A158447(n), n=1..100); # _Wesley Ivan Hurt_, Apr 26 2017
%t A158447 Table[10n^2-1,{n,50}]
%t A158447 LinearRecurrence[{3,-3,1},{9,39,89},50] (* _Harvey P. Dale_, Dec 08 2017 *)
%o A158447 (Magma) I:=[9, 39, 89]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
%o A158447 (PARI) a(n) = 10*n^2 - 1.
%Y A158447 Cf. A005843, A033583, A085787, A158446.
%K A158447 nonn,easy
%O A158447 1,1
%A A158447 _Vincenzo Librandi_, Mar 19 2009