A322595 -id:A322595

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A322594

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

Table of n, a(n) for n=0..41.

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).

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).

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Franck Maminirina Ramaharo, Dec 18 2018

STATUS

approved

A321124

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

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..41.

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.

Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.

L. N. Dobrodeev, Cubature formulas of the seventh order of accuracy for a hypersphere and a hypercube, USSR Computational Mathematics and Mathematical Physics Vol. 10 (1970), 252-253.

G. M. Phillips, Numerical integration over an N-dimensional rectangular region, The Computer Journal Vol. 10 (1967), 297-299.

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

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

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Franck Maminirina Ramaharo, Dec 17 2018

STATUS

approved

A322597

a(n) = (4*n^3 - 6*n^2 + 20*n + 3)/3.

+10
0

1, 7, 17, 39, 81, 151, 257, 407, 609, 871, 1201, 1607, 2097, 2679, 3361, 4151, 5057, 6087, 7249, 8551, 10001, 11607, 13377, 15319, 17441, 19751, 22257, 24967, 27889, 31031, 34401, 38007, 41857, 45959, 50321, 54951, 59857, 65047, 70529, 76311, 82401, 88807

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

For n >= 2, a(n) gives the number of function evaluations for Dooren and Ridder's degree 5 and 7 cubature rule over an n-dimensional cube, with the exception of a(3) = 45 and a(4) = 97.

LINKS

Table of n, a(n) for n=0..41.

Ronald Cools, Encyclopaedia of Cubature Formulas

Paul van Dooren and Luc de Ridder, An adaptive algorithm for numerical integration over an n-dimensional cube, Journal of Computational and Applied Mathematics, Vol. 2 (1976), 207-217.

Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics, Vol. 6 (1980), 295-302.

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.

G.f.: (1 + 3*x - 5*x^2 + 9*x^3)/((1 - x)^4).

E.g.f.: (1/3)*(3 + 18*x + 6*x^2 + 4*x^3)*exp(x).

MAPLE

[(4*n^3-6*n^2+20*n+3)/3$n=0..50]; # Muniru A Asiru, Jan 23 2019