A322595 - OEIS

A322595

a(n) = (n^3 + 9*n + 14*n + 9)/3.

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

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OFFSET

0,1

COMMENTS

For n >= 6, a(n) is the number of evaluating points on the hypersphere in R^n in Stoyanovas's degree 7 cubature rule.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Ronald Cools, Encyclopaedia of Cubature Formulas

Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

Srebra B. Stoyanova, Cubature of the seventh degree of accuracy for the hypersphere, Journal of Computational and Applied Mathematics Vol. 84 (1997), 15-21.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

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

a(n) = 2*binomial(n + 1, 3) + 6*binomial(n + 1, 2) + 2*binomial(n + 1, 1) + 1.

G.f.: (3 - x - 5*x^2 + 5*x^3)/(1 - x)^4. [Corrected by Georg Fischer, May 23 2019]

E.g.f.: (1/3)*(9 + 24*x + 12*x^2 + x^3)*exp(x).

MATHEMATICA

Table[(n^3 + 9*n + 14*n + 9)/3, {n, 0, 50}]

LinearRecurrence[{4, -6, 4, -1}, {3, 11, 21, 35}, 50] (* Harvey P. Dale, Aug 19 2020 *)

PROG

(Maxima) makelist((n^3 + 9*n + 14*n + 9)/3, n, 0, 50);

(Magma) [(n^3 + 9*n + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019