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%I #56 Mar 30 2024 02:43:52
%S 1,11,45,119,249,451,741,1135,1649,2299,3101,4071,5225,6579,8149,9951,
%T 12001,14315,16909,19799,23001,26531,30405,34639,39249,44251,49661,
%U 55495,61769,68499,75701,83391,91585,100299,109549,119351,129721,140675,152229,164399
%N a(n) = (2*n+1)*(4*n^2+4*n+3)/3.
%C For n>0, 30*a(n) is the sum of the ten distinct products of 2*n-1, 2*n+1, and 2*n+3. For example, when n = 1, we sum the ten distinct products of 1, 3, and 5: 1*1*1 + 1*1*3 + 1*1*5 + 1*3*3 + 1*3*5 + 1*5*5 + 3*3*3 + 3*3*5 + 3*5*5 + 5*5*5 = 330 = 30*11 = 30*a(1). - _J. M. Bergot_, Apr 06 2014
%H Vincenzo Librandi, <a href="/A057813/b057813.txt">Table of n, a(n) for n = 0..1000</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (10).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 2*A050533(n) + 1. - _N. J. A. Sloane_, Sep 22 2004
%F G.f.: (1+7*x+7*x^2+x^3)/(1-x)^4. - _Colin Barker_, Mar 01 2012
%F G.f. for sequence with interpolated zeros: 1/(8*x)*sinh(8*arctanh(x)) = 1/(16*x)*( ((1 + x)/(1 - x))^4 - ((1 - x)/(1 + x))^4 ) = 1 + 11*x^2 + 45*x^4 + 119*x^6 + .... Cf. A019560. - _Peter Bala_, Apr 07 2017
%F E.g.f.: (3 + 30*x + 36*x^2 + 8*x^3)*exp(x)/3. - _G. C. Greubel_, Dec 01 2017
%F From _Peter Bala_, Mar 26 2024: (Start)
%F 12*a(n) = (2*n + 1)*(a(n + 1) - a(n - 1)).
%F Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 3*Pi/16 - 1/2. Cf. A016754 and A336266. (End)
%p A057813:=n->(2*n + 1)*(4*n^2 + 4*n + 3)/3; seq(A057813(n), n=0..50); # _Wesley Ivan Hurt_, Apr 06 2014
%t Table[(2*n + 1)*(4*n^2 + 4*n + 3)/3, {n, 0, 50}] (* _David Nacin_, Mar 01 2012 *)
%o (PARI) P(x, y, z) = x^3 + x^2*y + x^2*z + x*y^2 + x*y*z + x*z^2 + y^3 + y^2*z + y*z^2 + z^3;
%o a(n) = P(2*n-1, 2*n+1, 2*n+3)/30; \\ _Michel Marcus_, Apr 22 2014
%o (Magma) [(2*n+1)*(4*n^2+4*n+3)/3 : n in [0..50]] // _Wesley Ivan Hurt_, Apr 22 2014
%Y 1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
%Y Cf. A019560, A336266.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 07 2000