|
%I #73 Jan 08 2023 02:39:22
%S -1,3,9,17,27,39,53,69,87,107,129,153,179,207,237,269,303,339,377,417,
%T 459,503,549,597,647,699,753,809,867,927,989,1053,1119,1187,1257,1329,
%U 1403,1479,1557,1637,1719,1803,1889,1977,2067,2159,2253,2349,2447,2547,2649
%N a(n) = n^2 + 3*n - 1.
%C Difference between n-th centered hexagonal number and (2n)^2. - _Alonso del Arte_, Jul 06 2004
%C Given the roots to n^2 + 3n - 1, a = -3.302775..., b = 0.302775...; then a(n) = (n + 3 + a) * (n + 3 + b). Example: a(3) = 17 = (6 - 3.302...) * (6 + 0.302775) - _Gary W. Adamson_, Jul 29 2009
%C a(n-1) = n*(n+1) - 3, with a(-1) = -3, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 13 for b = 2*n + 1. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 15 2013
%C Numbers m >= -1 such that 4m+13 is a square. - _Bruce J. Nicholson_, Jul 17 2017
%H Vincenzo Librandi, <a href="/A014209/b014209.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Centered_hexagonal_number">Centered Hexagonal Numbers</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F For n>0: a(n) = A176271(n+1,n). - _Reinhard Zumkeller_, Apr 13 2010
%F a(n) = a(n-1)+2*n+2, with n>0, a(0)=-1. - _Vincenzo Librandi_, Nov 20 2010
%F From _Colin Barker_, Feb 12 2012: (Start)
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
%F G.f.: (-1+6*x-3*x^2)/(1-x)^3. (End)
%F Sum_{n>=0} 1/a(n) = 1/3 + tan(sqrt(13)*Pi/2)*Pi/sqrt(13). - _Amiram Eldar_, Jan 08 2023
%t Table[n^2+3*n-1, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Oct 08 2009 *)
%t CoefficientList[Series[(- 1 + 6 x - 3 x^2)/(1 - x)^3, {x, 0, 60}], x] (* _Vincenzo Librandi_, Oct 15 2013 *)
%o (PARI) a(n)=n^2+3*n-1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A002522, A003215, A176271.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
|
