login
a(n) = n^2 + n + 6.
7

%I #35 Oct 28 2024 18:49:19

%S 6,8,12,18,26,36,48,62,78,96,116,138,162,188,216,246,278,312,348,386,

%T 426,468,512,558,606,656,708,762,818,876,936,998,1062,1128,1196,1266,

%U 1338,1412,1488,1566,1646,1728,1812,1898,1986,2076,2168,2262,2358,2456,2556

%N a(n) = n^2 + n + 6.

%H Muniru A Asiru, <a href="/A027691/b027691.txt">Table of n, a(n) for n = 0..2000</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)=6. - _Vincenzo Librandi_, Aug 05 2010

%F Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(23)/2)/sqrt(23). - _Amiram Eldar_, Jan 17 2021

%F From _Elmo R. Oliveira_, Oct 28 2024: (Start)

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

%F E.g.f.: (2*(3 + x) + x^2)*exp(x).

%F a(n) = 2*A152950(n+1). (End)

%t Table[n^2+n+6,{n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 12 2011 *)

%o (PARI) a(n)=n^2+n+6 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (GAP) List([0..50],n->n^2+n+6); # _Muniru A Asiru_, Jul 15 2018

%Y Cf. A002061, A002378, A002522, A152950.

%K nonn,easy

%O 0,1

%A _Patrick De Geest_

%E Offset changed by _Charles R Greathouse IV_, Jul 25 2010