A322595 -id:A322595 - cp's OEIS frontend

A322594 a(n) = (4n^3 + 12n^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

Franck Maminirina Ramaharo, Dec 18 2018

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

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

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).

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

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

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

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).

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

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

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).

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

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

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

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.

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Jan 23 2019

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.

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).

First differences: 2*A093328.

Cf. A174794, A321124, A322594, A322595.

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

Table[(4*n^3 - 6*n^2 + 20*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 - 6*n^2 + 20*n + 3)/3, n, 0, 50);

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).

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

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A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

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A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.

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cp's OEIS Frontend

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

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Mathematica

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Formula

A321124 a(n) = (4*n^3 - 6*n^2 + 14*n + 3)/3.

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Mathematica

Maxima

Formula

A322597 a(n) = (4*n^3 - 6*n^2 + 20*n + 3)/3.

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Maple

Mathematica

Maxima

Formula

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

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

A322597 a(n) = (4n^3 - 6n^2 + 20*n + 3)/3.