reviewed

approved

proposed

reviewed

editing

proposed

Rudolf M. Schürer, High-Dimensional Numerical Integration on Parallel Computers, Phd Dissertation, 2001. p. 80.

Rudolf M. Schürer, <a href="https://pdfs.semanticscholar.org/9d10/b4c1dc2118cce7865a193660be05e90d145a.pdf">High-Dimensional Numerical Integration on Parallel Computers</a>, Phd Dissertation, 2001. p. 80.

approved

editing

proposed

approved

editing

proposed

Ronald Cools, <a href="https://doi.org/10.1016/S0377-0427(99)00229-0">Monomial cubature rules since "Stroud": a compilation - part 2, </a>, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

Ronald Cools, <a href="https://doi.org/10.1016/S0377-0427(99)00229-0">Monomial cubature rules since “"Stroud”": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

For n >= 5, a(n) is the number of evaluation points on the n-dimensional cube in Phillips-DobrodevDobrodeev's degree 7 cubature rule.

Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/ndim_C.html">Encyclopaedia of Cubature formulae for the n-dimensional cubeFormulas</a>

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

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

(Maxima) makelist((4*n^3 - 6*n^2 + 14*n + 3)/3, n, 0, 50);

Cf. A000292, A128445, A161680, A174794, A322594, A322595.

allocated for Franck Maminirina Ramaharo

a(n) = (4*n^3 - 6*n^2 + 14*n + 3)/3.

1, 5, 13, 33, 73, 141, 245, 393, 593, 853, 1181, 1585, 2073, 2653, 3333, 4121, 5025, 6053, 7213, 8513, 9961, 11565, 13333, 15273, 17393, 19701, 22205, 24913, 27833, 30973, 34341, 37945, 41793, 45893, 50253, 54881, 59785, 64973, 70453, 76233, 82321, 88725

0,2

For n >= 5, a(n) is the number of evaluation points on the n-dimensional cube in Phillips-Dobrodev's degree 7 cubature rule.

Rudolf M. Schürer, High-Dimensional Numerical Integration on Parallel Computers, Phd Dissertation, 2001. p. 80.

Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/ndim_C.html">Cubature formulae for the n-dimensional cube</a>

Ronald Cools, <a href="https://doi.org/10.1016/S0377-0427(99)00229-0">Monomial cubature rules since “Stroud”: a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.

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.

L. N. Dobrodeev, <a href="https://doi.org/10.1016/0041-5553(70)90084-4">Cubature formulas of the seventh order of accuracy for a hypersphere and a hypercube</a>, USSR Computational Mathematics and Mathematical Physics Vol. 10 (1970), 252-253.

G. M. Phillips, <a href="https://doi.org/10.1093/comjnl/10.3.297">Numerical integration over an N-dimensional rectangular region</a>, The Computer Journal Vol. 10 (1967), 297-299.

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

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

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

a(n) = a(n-1) + A128445(n+1), n >= 1.

E.g.f.: (1/3)*(3 + 12*x + 6*x^2 + 4*x^3)*exp(x).

G.f.: (1 + x - x^2 + 7*x^3)/(1 - x)^4.

Table[(4*n^3 - 6*n^2 + 14*n + 3)/3, {n, 0, 50}]

(Maxima) makelist((4*n^3 - 6*n^2 + 14*n + 3)/3, n, 0, 50);

allocated

nonn,easy

Franck Maminirina Ramaharo, Dec 17 2018

approved

editing