A117951 - OEIS

%I #56 Feb 16 2025 08:33:00

%S 5,6,9,14,21,30,41,54,69,86,105,126,149,174,201,230,261,294,329,366,

%T 405,446,489,534,581,630,681,734,789,846,905,966,1029,1094,1161,1230,

%U 1301,1374,1449,1526,1605,1686,1769,1854,1941,2030,2121,2214,2309,2406,2505

%N a(n) = n^2 + 5.

%H Ivan Panchenko, <a href="/A117951/b117951.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.

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

%F a(n) = 2*n + a(n-1) - 1 (with a(0)=5). - _Vincenzo Librandi_, Nov 13 2010

%F From _Colin Barker_, Apr 10 2012: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: (5-9*x+6*x^2)/(1-x)^3. (End)

%F From _Amiram Eldar_, Jul 13 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + sqrt(5)*Pi*coth(sqrt(5)*Pi))/10.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(5)*Pi*cosech(sqrt(5)*Pi))/10. (End)

%F From _Amiram Eldar_, Feb 05 2024: (Start)

%F Product_{n>=0} (1 - 1/a(n)) = 2*sinh(2*Pi)/(sqrt(5)*sinh(sqrt(5)*Pi)).

%F Product_{n>=0} (1 + 1/a(n)) = sqrt(6/5)*sinh(sqrt(6)*Pi)/sinh(sqrt(5)*Pi). (End)

%F E.g.f.: exp(x)*(5 + x + x^2). - _Elmo R. Oliveira_, Jan 17 2025

%t Range[0,50]^2+5 (* or *) LinearRecurrence[{3,-3,1},{5,6,9},60] (* _Harvey P. Dale_, Aug 04 2020 *)

%o (SageMath) [lucas_number1(3,n,-5) for n in range(0, 51)] # _Zerinvary Lajos_, May 16 2009

%o (PARI) a(n)=n^2+5 \\ _Charles R Greathouse IV_, Apr 10 2012

%Y Cf. A078402.

%K nonn,easy

%O 0,1

%A _Eric W. Weisstein_, Apr 04 2006