A322594 - OEIS

%I #7 Dec 29 2018 23:53:14

%S 1,5,25,69,145,261,425,645,929,1285,1721,2245,2865,3589,4425,5381,

%T 6465,7685,9049,10565,12241,14085,16105,18309,20705,23301,26105,29125,

%U 32369,35845,39561,43525,47745,52229,56985,62021,67345,72965,78889,85125,91681,98565

%N a(n) = (4*n^3 + 12*n^2 - 4*n + 3)/3.

%C a(n) is the number of evaluation points on the n-dimensional cube in Lyness's degree 7 cubature rule.

%D Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

%H Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/">Encyclopaedia of Cubature Formulas</a>

%H Ronald Cools and Philip Rabinowitz, <a href="https://doi.org/10.1016/0377-0427(93)90027-9">Monomial cubature rules since "Stroud": a compilation</a>, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.

%H James Lu and David L. Darmofal, <a href="https://doi.org/10.1137/S1064827503426863">Higher-dimensional integration with gaussian weight for applications in probabilistic design</a>, SIAM J. Sci. Comput. Vol. 26 (2004), 613-624.

%H James N. Lyness, <a href="https://dx.doi.org/10.2307/2003669">Symmetric integration rules for hypercubes II. Rule projection and rule extension</a>, Math. Comp. Vol. 19 (1965), 394-407.

%H James N. Lyness, <a href="https://doi.org/10.1090/S0025-5718-1965-0201070-3">Integration rules of hypercubic symmetry over a certain spherically symmetric space</a>, Math. Comp. Vol. 19 (1965), 471-476.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 5.

%F a(n) = a(n-1) + 4*A028387(n-1), n >= 1.

%F a(n) = 8*binomial(n, 3) + 16*binomial(n, 2) + 4*binomial(n, 1) + 1.

%F G.f.: (1 + x + 11*x^2 - 5*x^3)/(1 - x)^4

%F E.g.f.: (1/3)*(3 + 12*x + 24*x^2 + 4*x^3)*exp(x).

%t Table[(4*n^3 + 12*n^2 - 4*n + 3)/3, {n, 0, 50}]

%o (Maxima) makelist((4*n^3 + 12*n^2 - 4*n + 3)/3, n, 0, 50);

%Y Cf. A000292, A161680, A174794, A321124, A322595.

%K nonn,easy

%O 0,2

%A _Franck Maminirina Ramaharo_, Dec 18 2018