A166148 - OEIS

%I #30 Mar 17 2024 02:12:59

%S 1,19,46,82,127,181,244,316,397,487,586,694,811,937,1072,1216,1369,

%T 1531,1702,1882,2071,2269,2476,2692,2917,3151,3394,3646,3907,4177,

%U 4456,4744,5041,5347,5662,5986,6319,6661,7012,7372,7741,8119,8506,8902,9307,9721,10144

%N a(n) = (9*n^2 + 9*n - 16)/2.

%H Vincenzo Librandi, <a href="/A166148/b166148.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(n) = a(n-1) + 9*n with n > 1, a(1)=1.

%F From _Vincenzo Librandi_, Mar 15 2012: (Start)

%F G.f.: x*(1+16*x-8*x^2)/(1-x)^3.

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

%F E.g.f.: (1/2)*(9*x^2 + 18*x - 16)*exp(x). - _G. C. Greubel_, May 01 2016

%F Sum_{n>=1} 1/a(n) = 1/8 + (2*Pi/(3*sqrt(73)))*tan(sqrt(73)*Pi/6). - _Amiram Eldar_, Feb 20 2023

%t CoefficientList[Series[(1+16x-8x^2)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 19, 46}, 50] (* _Vincenzo Librandi_, Mar 15 2012 *)

%o (PARI) a(n)=9*binomial(n+1,2)-8 \\ _Charles R Greathouse IV_, Jan 11 2012

%o (Magma) I:=[1, 19, 46]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Mar 15 2012

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Oct 08 2009

%E New name from _Charles R Greathouse IV_, Jan 11 2012