A321123 a(n) = 2^n + 2*n^2 + 2*n + 1.

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

Offset: 0

Author

Franck Maminirina Ramaharo, Dec 17 2018

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is the number of evaluation points on the n-dimensional cube in Genz and Malik's degree 7 cubature rule.

Links

• Marius A. Burtea, Table of n, a(n) for n = 0..201
• 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.
• 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 (5,-9,7,-2).

Crossrefs

Cf. A000051, A000079, A001787, A001844, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A322593.

Programs

Magma

[2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Mathematica

Table[2^n + 2*n^2 + 2*n + 1, {n, 0, 50}]

Maxima

makelist(2^n + 2*n^2 + 2*n + 1, n, 0, 50);

Formula

a(n) = A000079(n) + A001844(n).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), n >= 4.

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

E.g.f.: exp(2*x) + (1 + 4*x + 2*x^2)*exp(x).