A322594 a(n) = (4*n^3 + 12*n^2 - 4*n + 3)/3.

1, 5, 25, 69, 145, 261, 425, 645, 929, 1285, 1721, 2245, 2865, 3589, 4425, 5381, 6465, 7685, 9049, 10565, 12241, 14085, 16105, 18309, 20705, 23301, 26105, 29125, 32369, 35845, 39561, 43525, 47745, 52229, 56985, 62021, 67345, 72965, 78889, 85125, 91681, 98565

Offset: 0

Author

Franck Maminirina Ramaharo, Dec 18 2018

Keywords

• nonn
• easy

Comments

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

References

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

Links

• Ronald Cools, Encyclopaedia of Cubature Formulas
• Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
• James Lu and David L. Darmofal, Higher-dimensional integration with gaussian weight for applications in probabilistic design, SIAM J. Sci. Comput. Vol. 26 (2004), 613-624.
• James N. Lyness, Symmetric integration rules for hypercubes II. Rule projection and rule extension, Math. Comp. Vol. 19 (1965), 394-407.
• James N. Lyness, Integration rules of hypercubic symmetry over a certain spherically symmetric space, Math. Comp. Vol. 19 (1965), 471-476.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

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

Programs

Mathematica

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

Maxima

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

Formula

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

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

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

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

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