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A272039 a(n) = 10*n^2 + 4*n + 1.

%I A272039 #58 Sep 08 2022 08:46:16
%S A272039 1,15,49,103,177,271,385,519,673,847,1041,1255,1489,1743,2017,2311,
%T A272039 2625,2959,3313,3687,4081,4495,4929,5383,5857,6351,6865,7399,7953,
%U A272039 8527,9121,9735,10369,11023,11697,12391,13105,13839,14593,15367,16161,16975,17809,18663,19537
%N A272039 a(n) = 10*n^2 + 4*n + 1.
%C A272039 Polynomials from the table "Coefficients and roots of Ehrhart polynomials" in Beck et al. paper (see Links section):
%C A272039 . Cube:                   A000578;
%C A272039 . Cube minus corner:      A004068;
%C A272039 . Prism:                  A002411;
%C A272039 . Octahedron:             A005900;
%C A272039 . Square pyramid:         A000330;
%C A272039 . Bypyramid:              A006003;
%C A272039 . Unimodular tetrahedron: A000292;
%C A272039 . Fat tetrahedron:        A167875;
%C A272039 . Cyclic(2,5), which has the same polynomial form of this sequence.
%C A272039 a(n) for n = 0, -1, 1, -2, 2, -3, 3, ... gives all x such that (5*x - 3)/2 is a square.
%C A272039 Squares in sequence: 1, 49, 1385329, 101263969, 2880599856289, ...
%C A272039 Is this 1 followed by A228219?
%H A272039 M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Coefficients and roots of Ehrhart Polynomials</a>, page 19.
%H A272039 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A272039 O.g.f.: (1 + 12*x + 7*x^2)/(1 - x)^3.
%F A272039 E.g.f.: (1 + 14*x + 10*x^2)*exp(x).
%F A272039 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A272039 a(n) = 2*A168668(n) + 1.
%t A272039 Table[10 n^2 + 4 n + 1, {n, 0, 50}]
%t A272039 LinearRecurrence[{3,-3,1},{1,15,49},50] (* _Harvey P. Dale_, Dec 26 2021 *)
%o A272039 (Magma) [10*n^2+4*n+1: n in [0..50]];
%o A272039 (PARI) a(n)=10*n^2+4*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A272039 Cf. A000292, A000330, A000578, A002411, A004068, A005900, A006003, A167875, A168668, A228219, A272124, A272130.
%K A272039 nonn,easy
%O A272039 0,2
%A A272039 _Vincenzo Librandi_, Apr 20 2016
%E A272039 Edited by _Bruno Berselli_, Apr 22 2016