A322595 - OEIS

A322595

a(n) = (n^3 + 9*n^2 + 14*n + 9)/3.

3, 11, 27, 53, 91, 143, 211, 297, 403, 531, 683, 861, 1067, 1303, 1571, 1873, 2211, 2587, 3003, 3461, 3963, 4511, 5107, 5753, 6451, 7203, 8011, 8877, 9803, 10791, 11843, 12961, 14147, 15403, 16731, 18133, 19611, 21167, 22803, 24521, 26323, 28211, 30187, 32253

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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

Stefano Spezia, Table of n, a(n) for n = 0..10000

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 + x^2 - x^3)/(1 - x)^4.

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

MATHEMATICA

Table[(n^3 + 9*n^2 + 14*n + 9)/3, {n, 0, 50}]

LinearRecurrence[{4, -6, 4, -1}, {3, 11, 27, 53}, 50] (* Harvey P. Dale, Aug 19 2020 *)

PROG

(Maxima) makelist((n^3 + 9*n^2 + 14*n + 9)/3, n, 0, 50);

(Magma) [(n^3 + 9*n^2 + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019