A166150 - OEIS

%I #35 Mar 17 2024 02:13:04

%S 1,21,51,91,141,201,271,351,441,541,651,771,901,1041,1191,1351,1521,

%T 1701,1891,2091,2301,2521,2751,2991,3241,3501,3771,4051,4341,4641,

%U 4951,5271,5601,5941,6291,6651,7021,7401,7791,8191,8601,9021,9451,9891,10341

%N a(n) = 5*n^2 + 5*n - 9.

%C First differences are in A008592.

%H Vincenzo Librandi, <a href="/A166150/b166150.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) + 10*n (with a(1)=1).

%F G.f.: x*(1+18*x-9*x^2)/(1-x)^3. - _Vincenzo Librandi_, Sep 13 2013

%F From _G. C. Greubel_, May 01 2016: (Start)

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

%F E.g.f.: (5*x^2 + 10*x - 9)*exp(x) + 9. (End)

%F Sum_{n>=1} 1/a(n) = 1/9 + (Pi/sqrt(205))*tan(sqrt(41/5)*Pi/2). - _Amiram Eldar_, Feb 20 2023

%p A166150:=n->5*n^2+5*n-9: seq(A166150(n), n=1..100); # _Wesley Ivan Hurt_, May 01 2016

%t Table[(5 n^2 + 5  n - 9), {n, 50}] (* or *) CoefficientList[Series[(1 + 18 x - 9 x^2)/(1 - x)^3, {x, 0, 45}], x] (* _Vincenzo Librandi_, Sep 13 2013 *)

%t LinearRecurrence[{3,-3,1},{1,21,51}, 50] (* _G. C. Greubel_, May 01 2016 *)

%o (PARI) a(n)=5*n*(n+1)-9 \\ _Charles R Greathouse IV_, Jan 11 2012

%o (Magma) [5*n^2+5*n-9: n in [1..45]]; // _Vincenzo Librandi_, Sep 13 2013

%Y Cf. A008592.

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Oct 08 2009

%E a(29)-a(45) corrected by _Charles R Greathouse IV_, Jan 11 2012

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