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A217873 a(n) = 4*n*(n^2 + 2)/3.

%I A217873 #36 Aug 14 2025 01:09:09
%S A217873 0,4,16,44,96,180,304,476,704,996,1360,1804,2336,2964,3696,4540,5504,
%T A217873 6596,7824,9196,10720,12404,14256,16284,18496,20900,23504,26316,29344,
%U A217873 32596,36080,39804,43776,48004,52496,57260,62304,67636,73264,79196,85440,92004
%N A217873 a(n) = 4*n*(n^2 + 2)/3.
%C A217873 Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.
%H A217873 Vincenzo Librandi, <a href="/A217873/b217873.txt">Table of n, a(n) for n = 0..1000</a>
%H A217873 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A217873 a(n) = 4*A006527(n).
%F A217873 From _Peter Luschny_, Oct 14 2012: (Start)
%F A217873 a(n) = A008412(n)/2.
%F A217873 a(n) = A174794(n+1) - 1.
%F A217873 First differences are in A112087.
%F A217873 Second differences are in A008590 and A022144.
%F A217873 Binomial transformation of (a(n), n > 0) is A082138.  (End)
%F A217873 G.f.: 4*x*(1 + x^2)/(x - 1)^4. - _R. J. Mathar_, Oct 15 2012
%F A217873 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - _Harvey P. Dale_, Mar 16 2015
%F A217873 From _Elmo R. Oliveira_, Aug 09 2025: (Start)
%F A217873 E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
%F A217873 a(n) = A292022(n)/3. (End)
%t A217873 Table[4n(n^2 + 2)/3, {n, 0, 39}] (* _Alonso del Arte_, Oct 22 2012 *)
%t A217873 LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* _Harvey P. Dale_, Mar 16 2015 *)
%o A217873 (PARI) a(n)=(n^2+2)*n/3*4
%o A217873 (Maxima) makelist(4*n*(n^2+2)/3, n, 0, 41); /* _Martin Ettl_, Oct 15 2012 */
%o A217873 (Magma) [4*n*(n^2+2)/3: n in [0..45]]; // _Vincenzo Librandi_, Nov 08 2012
%Y A217873 Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.
%K A217873 nonn,easy
%O A217873 0,2
%A A217873 _M. F. Hasler_, Oct 13 2012