A052460 - OEIS

%I #33 Feb 16 2025 08:32:42

%S 1,50,675,4624,21125,73926,214375,540800,1225449,2550250,4952651,

%T 9082800,15873325,26622974,43095375,67634176,103295825,154001250,

%U 224707699,321602000,452316501,626168950,854427575,1150602624

%N 3-magic series constant.

%H Kelvin Voskuijl, <a href="/A052460/b052460.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MagicConstant.html">Magic Constant</a>.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

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

%F G.f.: x*(1 +42*x +303*x^2 +568*x^3 +303*x^4 +42*x^5 +x^6)/(1-x)^8 .

%F a(1)=1, a(2)=50, a(3)=675, a(4)=4624, a(5)=21125, a(6)=73926, a(7)= 214375, a(8)=540800, a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) + 56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). - _Harvey P. Dale_, Aug 14 2013

%F E.g.f.: x*(4 +96*x +352*x^2 +370*x^3 +142*x^4 +21*x^5 + x^6)*exp(x)/4. - _G. C. Greubel_, Sep 23 2019

%p seq(n^3*(1+n^2)^2/4, n=1..30); # _G. C. Greubel_, Sep 23 2019

%t Table[n^3/4 (n^2+1)^2,{n,30}] (* or *) LinearRecurrence[{8,-28,56,-70,56, -28,8,-1}, {1,50,675,4624,21125,73926,214375,540800}, 30] (* _Harvey P. Dale_, Aug 14 2013 *)

%t CoefficientList[Series[(1+42x+303x^2+568x^3+303x^4+42x^5+x^6)/(1-x)^8, {x,0,40}], x] (* _Vincenzo Librandi_, Aug 14 2013 *)

%o (PARI) vector(30, n, n^3*(1+n^2)^2/4) \\ _G. C. Greubel_, Sep 23 2019

%o (Magma) [n^3*(1+n^2)^2/4: n in [1..30]]; // _G. C. Greubel_, Sep 23 2019

%o (Sage) [n^3*(1+n^2)^2/4 for n in (1..30)] # _G. C. Greubel_, Sep 23 2019

%o (GAP) List([1..30], n-> n^3*(1+n^2)^2/4); # _G. C. Greubel_, Sep 23 2019

%Y Cf. A052459, A052461.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_