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a(n) = 4*n^2 - 6*n + 1.
15

%I #53 Apr 22 2024 06:25:58

%S -1,5,19,41,71,109,155,209,271,341,419,505,599,701,811,929,1055,1189,

%T 1331,1481,1639,1805,1979,2161,2351,2549,2755,2969,3191,3421,3659,

%U 3905,4159,4421,4691,4969,5255,5549,5851,6161,6479,6805,7139,7481,7831,8189,8555,8929,9311,9701,10099,10505,10919,11341

%N a(n) = 4*n^2 - 6*n + 1.

%H Vincenzo Librandi, <a href="/A125202/b125202.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A125199(n,n-1) for n>1.

%F A003415(a(n)) = A017089(n-1).

%F From _Arkadiusz Wesolowski_, Dec 25 2011: (Start)

%F a(1) = -1, a(n) = a(n-1) + 8*n - 10.

%F a(n) = 2*a(n-1) - a(n-2) + 8 with a(1) = -1 and a(2) = 5.

%F G.f.: (1 - 4*x + 11*x^2)/(1 - x)^3. (End)

%F a(n) = A002943(n-1) - 1. - _Arkadiusz Wesolowski_, Feb 15 2012

%F a(n) = A028387(2n-3), with A028387(-1) = -1. - _Vincenzo Librandi_, Oct 10 2013

%F E.g.f.: exp(x)*(1 - 2*x + 4*x^2). - _Stefano Spezia_, Oct 10 2022

%F Sum_{n>=1} 1/a(n) = sqrt(5)/10*(psi(1/4+sqrt(5)/4) - psi(1/4-sqrt(5)/4)) = -0.656213833... - _R. J. Mathar_, Apr 22 2024

%t f[a_]:=4*a^2-6*a+1; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 14 2009 *)

%o (Magma) [4*n^2 - 6*n + 1: n in [1..60]]; // _Vincenzo Librandi_, Jul 11 2011

%o (PARI) a(n)=4*n^2-6*n+1 \\ _Charles R Greathouse IV_, Sep 28 2015

%Y Cf. A125199.

%Y Cf. A003415, A017089.

%Y Cf. A002939, A016789, A017041, A017485, A028387.

%Y Cf. A002943, A028387.

%K sign,easy

%O 1,2

%A _Reinhard Zumkeller_, Nov 24 2006