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A008511 Number of points on surface of 4-dimensional cube.

%I A008511 #43 Sep 08 2022 08:44:35
%S A008511 0,16,80,240,544,1040,1776,2800,4160,5904,8080,10736,13920,17680,
%T A008511 22064,27120,32896,39440,46800,55024,64160,74256,85360,97520,110784,
%U A008511 125200,140816,157680,175840
%N A008511 Number of points on surface of 4-dimensional cube.
%H A008511 Vincenzo Librandi, <a href="/A008511/b008511.txt">Table of n, a(n) for n = 0..10000</a>
%H A008511 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A008511 a(n) = (n+1)^4 - (n-1)^4 = 8*n + 8*n^3.
%F A008511 G.f.: 16*x*(1+x+x^2)/(1-4*x+6*x^2-4*x^3+x^4). - _Colin Barker_, Jan 02 2012
%F A008511 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=16, a(2)=80, a(3)=240. - _Harvey P. Dale_, Oct 15 2012
%F A008511 a(n) = 16 * A006003(n). - _J. M. Bergot_, Jul 22 2013
%F A008511 For n > 0, a(n) = A005917(n) + A005917(n+1) = A000583(n+1) - A000583(n-1). - _Bruce J. Nicholson_, Jun 19 2018
%F A008511 a(n) = -a(-n) for all n in Z. - _Michael Somos_, Jun 24 2018
%F A008511 E.g.f.:  8*x*(2 +3*x +x^2)*exp(x). - _G. C. Greubel_, Nov 09 2019
%e A008511 G.f. = 16*x + 80*x^2 + 240*x^3 + 544*x^4 + 1040*x^5 + 1776*x^6 + 2800*x^7 + ... - _Michael Somos_, Jun 24 2018
%p A008511 seq(8*n*(1+n^2), n=0..30); # _G. C. Greubel_, Nov 09 2019
%t A008511 Last[#]-First[#]&/@Partition[Range[-1,30]^4,3,1] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,16,80,240},30] (* _Harvey P. Dale_, Oct 15 2012 *)
%o A008511 (Magma) [(n+1)^4-(n-1)^4: n in [0..30]]; // _Vincenzo Librandi_, Aug 27 2011
%o A008511 (PARI) vector(31, n, 8*(n-1)*(1+(n-1)^2)) \\ _G. C. Greubel_, Nov 09 2019
%o A008511 (Sage) [8*n*(1+n^2) for n in (0..30)] # _G. C. Greubel_, Nov 09 2019
%o A008511 (GAP) List([0..30], n-> 8*n*(1+n^2)); # _G. C. Greubel_, Nov 09 2019
%Y A008511 Cf. A000583, A005917, A006003.
%K A008511 nonn,easy
%O A008511 0,2
%A A008511 _N. J. A. Sloane_