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a(n) = n + (n+1)^2 + (n+2)^3.
8

%I #60 Aug 07 2022 07:53:56

%S 9,32,75,144,245,384,567,800,1089,1440,1859,2352,2925,3584,4335,5184,

%T 6137,7200,8379,9680,11109,12672,14375,16224,18225,20384,22707,25200,

%U 27869,30720,33759,36992,40425,44064,47915,51984,56277,60800

%N a(n) = n + (n+1)^2 + (n+2)^3.

%C Numbers n > 0 such that x^3 + 2*x^2 + n factors over the integers. - _James R. Buddenhagen_, Apr 19 2005

%H Vincenzo Librandi, <a href="/A027620/b027620.txt">Table of n, a(n) for n = 0..10000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/sumpower.htm">Palindromic Quasi_Under_Squares of the form n+(n+1)^2</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (n+1)*(n+3)^2. - _Zerinvary Lajos_, Sep 24 2006, corrected Dec 21 2010

%F G.f.: (9 - 4*x + x^2)/(1 - x)^4. - _R. J. Mathar_, Dec 21 2010

%F a(n) = coefficient of x^3 in the Maclaurin expansion of -1/((n+3)*x^2 + (n+3)*x + 1). - _Francesco Daddi_, Aug 04 2011

%F E.g.f.: (9 + 23*x + 10*x^2 + x^3)*exp(x). - _G. C. Greubel_, Aug 05 2022

%p [seq((n+3)^2*(n+1), n=0..40)]; # _Zerinvary Lajos_, Sep 24 2006

%t Table[n +(n+1)^2 +(n+2)^3, {n, 0, 40}] (* or *) LinearRecurrence[{4,-6,4,-1},{9,32,75,144}, 40] (* _Harvey P. Dale_, Feb 23 2021 *)

%o (Sage) [i+(i+1)^2+(i+2)^3 for i in range(0,38)] # _Zerinvary Lajos_, Jul 03 2008

%o (Magma) [n + (n+1)^2 + (n+2)^3: n in [0..40]]; // _Vincenzo Librandi_, Aug 05 2011

%o (Maxima) A027620(n):=n + (n+1)^2 + (n+2)^3$ makelist(A027620(n),n,0,15); /* _Martin Ettl_, Dec 13 2012 */

%o (PARI) a(n)=n+(n+1)^2+(n+2)^3 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000027, A027621, A027622, A028387.

%K nonn,easy

%O 0,1

%A _Patrick De Geest_