%I #34 Sep 08 2022 08:46:16

%S 1,8,25,58,113,196,313,470,673,928,1241,1618,2065,2588,3193,3886,4673,

%T 5560,6553,7658,8881,10228,11705,13318,15073,16976,19033,21250,23633,

%U 26188,28921,31838,34945,38248,41753,45466,49393,53540,57913,62518,67361,72448

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

%C Numbers of the type (m+1)^3 - (m-1)*m. Similar sequences are: A069778 with the closed form (m+1)^3 - m*(m+1), A152015 with (m+1)^3 - (m+1)*(m+2).

%H Vincenzo Librandi, <a href="/A270867/b270867.txt">Table of n, a(n) for n = 0..1000</a>

%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015 (page 19, 4th row; page 21, 3rd row).

%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 + 4*x - x^2 + 2*x^3)/(1 - x)^4.

%F E.g.f.: (1 + 7*x + 5*x^2 + x^3)*exp(x).

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

%F a(n) = -A270109(-n-1). - _Bruno Berselli_, Apr 01 2016

%F a(n+2) - 2*a(n+1) + a(n) = A016957(n+1). - _Wesley Ivan Hurt_, Apr 02 2016

%p A270867:=n->n^3+2*n^2+4*n+1: seq(A270867(n), n=0..100); # _Wesley Ivan Hurt_, Apr 01 2016

%t Table[n^3 + 2 n^2 + 4 n + 1, {n, 0, 40}]

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

%o (PARI) x='x+O('x^99); Vec((1+4*x-x^2+2*x^3)/(1-x)^4) \\ _Altug Alkan_, Apr 01 2016

%o (Python) for i in range(0,100):print(i**3+2*i**2+4*i+1) # _Soumil Mandal_, Apr 02 2016

%Y Cf. A069778, A016957, A152015, A270109.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Apr 01 2016