%I #42 Oct 31 2024 15:28:59

%S 8,10,14,20,28,38,50,64,80,98,118,140,164,190,218,248,280,314,350,388,

%T 428,470,514,560,608,658,710,764,820,878,938,1000,1064,1130,1198,1268,

%U 1340,1414,1490,1568,1648,1730,1814,1900,1988,2078,2170,2264,2360,2458,2558

%N a(n) = n^2 + n + 8.

%H Muniru A Asiru, <a href="/A027693/b027693.txt">Table of n, a(n) for n = 0..4000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/quasimor.htm">Palindromic Quasi_Over_Squares of the form n^2+(n+X)</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) (with a(0)=8). - _Vincenzo Librandi_, Aug 05 2010

%F From _Harvey P. Dale_, Dec 13 2011: (Start)

%F a(0)=8, a(1)=10, a(2)=14, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: (2*(7-4*x)*x-8)/(x-1)^3. (End)

%F Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(31)/2)/sqrt(31). - _Amiram Eldar_, Jan 17 2021

%F From _Elmo R. Oliveira_, Oct 31 2024: (Start)

%F E.g.f.: exp(x)*(8 + 2*x + x^2).

%F a(n) = 2*A145018(n+1). (End)

%p with(combinat): seq(fibonacci(3, n)+n+7, n=0..46); # _Zerinvary Lajos_, Jun 07 2008

%t f[n_]:=n^2+n+8;f[Range[0,100]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 12 2011 *)

%t LinearRecurrence[{3,-3,1},{8,10,14},60] (* _Harvey P. Dale_, Dec 13 2011 *)

%o (PARI) a(n)=n^2+(n+8) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (GAP) List([0..50],n->n^2+n+8); # _Muniru A Asiru_, Jul 15 2018

%Y Cf. A002061, A002378, A002522, A145018.

%K nonn,easy

%O 0,1

%A _Patrick De Geest_