A272039 - OEIS

%I #58 Sep 08 2022 08:46:16

%S 1,15,49,103,177,271,385,519,673,847,1041,1255,1489,1743,2017,2311,

%T 2625,2959,3313,3687,4081,4495,4929,5383,5857,6351,6865,7399,7953,

%U 8527,9121,9735,10369,11023,11697,12391,13105,13839,14593,15367,16161,16975,17809,18663,19537

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

%C Polynomials from the table "Coefficients and roots of Ehrhart polynomials" in Beck et al. paper (see Links section):

%C . Cube:                   A000578;

%C . Cube minus corner:      A004068;

%C . Prism:                  A002411;

%C . Octahedron:             A005900;

%C . Square pyramid:         A000330;

%C . Bypyramid:              A006003;

%C . Unimodular tetrahedron: A000292;

%C . Fat tetrahedron:        A167875;

%C . Cyclic(2,5), which has the same polynomial form of this sequence.

%C a(n) for n = 0, -1, 1, -2, 2, -3, 3, ... gives all x such that (5*x - 3)/2 is a square.

%C Squares in sequence: 1, 49, 1385329, 101263969, 2880599856289, ...

%C Is this 1 followed by A228219?

%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F O.g.f.: (1 + 12*x + 7*x^2)/(1 - x)^3.

%F E.g.f.: (1 + 14*x + 10*x^2)*exp(x).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = 2*A168668(n) + 1.

%t Table[10 n^2 + 4 n + 1, {n, 0, 50}]

%t LinearRecurrence[{3,-3,1},{1,15,49},50] (* _Harvey P. Dale_, Dec 26 2021 *)

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

%o (PARI) a(n)=10*n^2+4*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000292, A000330, A000578, A002411, A004068, A005900, A006003, A167875, A168668, A228219, A272124, A272130.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Apr 20 2016

%E Edited by _Bruno Berselli_, Apr 22 2016