A174794 -id:A174794 - cp's OEIS frontend

A217873 a(n) = 4n(n^2 + 2)/3.

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004

Offset: 0

M. F. Hasler, Oct 13 2012

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

[4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012

Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)

makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */

PARI
```
a(n)=(n^2+2)*n/3*4
```

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138.  (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

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

Original entry on oeis.org

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).

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Dec 19 2018

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

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).

First differences: A027693.

Cf. A000292, A161680, A174794, A321124, A322594.

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

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 *)

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

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).

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).

cp's OEIS Frontend

A217873 a(n) = 4*n*(n^2 + 2)/3.

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

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A322595 a(n) = (n^3 + 9*n + 14*n + 9)/3.

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Magma

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

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Mathematica

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

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A217873 a(n) = 4n(n^2 + 2)/3.

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

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

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

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