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A134538 a(n) = 5*n^2 - 1.

%I A134538 #27 Jun 06 2025 00:17:18
%S A134538 4,19,44,79,124,179,244,319,404,499,604,719,844,979,1124,1279,1444,
%T A134538 1619,1804,1999,2204,2419,2644,2879,3124,3379,3644,3919,4204,4499,
%U A134538 4804,5119,5444,5779,6124,6479,6844,7219,7604,7999,8404,8819,9244,9679,10124,10579,11044
%N A134538 a(n) = 5*n^2 - 1.
%C A134538 For k != 0, the quintic polynomials of the form x^5 + 5*(5*k^2 - 1)*x + 4*(5*k^2 - 1) have Galois group A5 (order 60) over rational numbers.
%H A134538 Vincenzo Librandi, <a href="/A134538/b134538.txt">Table of n, a(n) for n = 1..1000</a>
%H A134538 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A134538 G.f.: x*(-4-7*x+x^2)/(-1+x)^3. - _R. J. Mathar_, Nov 14 2007
%F A134538 From _Amiram Eldar_, Feb 04 2021: (Start)
%F A134538 Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(5))*cot(Pi/sqrt(5)))/2.
%F A134538 Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(5))*csc(Pi/sqrt(5)) - 1)/2.
%F A134538 Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(5))*csc(Pi/sqrt(5)).
%F A134538 Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(5))*sin(sqrt(2/5)*Pi)/sqrt(2). (End)
%F A134538 From _Elmo R. Oliveira_, Jun 04 2025: (Start)
%F A134538 E.g.f.: 1 + (-1 + 5*x + 5*x^2)*exp(x).
%F A134538 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%t A134538 Table[5n^2 - 1, {n, 1, 50}]
%t A134538 CoefficientList[Series[(4+7*x-x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 09 2012 *)
%o A134538 (Magma) [5*n^2-1: n in [1..50]]; // _Vincenzo Librandi_, Jul 09 2012
%o A134538 (PARI) a(n)=5*n^2-1 \\ _Charles R Greathouse IV_, Jul 01 2013
%K A134538 nonn,easy
%O A134538 1,1
%A A134538 _Artur Jasinski_, Oct 30 2007