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a(n) = 16*n^3 + 10*n^2 + 4*n + 1.
2

%I #16 Sep 08 2022 08:46:16

%S 1,31,177,535,1201,2271,3841,6007,8865,12511,17041,22551,29137,36895,

%T 45921,56311,68161,81567,96625,113431,132081,152671,175297,200055,

%U 227041,256351,288081,322327,359185,398751,441121,486391,534657,586015,640561,698391

%N a(n) = 16*n^3 + 10*n^2 + 4*n + 1.

%H Vincenzo Librandi, <a href="/A272130/b272130.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/ehrhart.pdf">Coefficients and roots of Ehrhart Polynomials</a>, Contemp. Math. 374 (2005), 15-36, page 19.

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

%F O.g.f.: (1+27*x+59*x^2+9*x^3)/(1-x)^4.

%F E.g.f.: (1+30*x+58*x^2+16*x^3)*exp(x).

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.

%F a(n) = A158187(n) + A144965(n).

%p A272130:=n->16*n^3+10*n^2+4*n+1: seq(A272130(n), n=0..50); # _Wesley Ivan Hurt_, Apr 22 2016

%t LinearRecurrence[{4,-6,4,-1},{1,31,177,535},50]

%t CoefficientList[Series[(1 + 27*x + 59*x^2 + 9*x^3)/(1 - x)^4, {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Apr 22 2016 *)

%o (Magma) [16*n^3+10*n^2+4*n+1: n in [0..50]];

%o (PARI) vector(100, n, n--; 16*n^3+10*n^2+4*n+1) \\ _Altug Alkan_, Apr 22 2016

%Y Cf. A144965, A158187, A272039.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Apr 21 2016