A321123 - OEIS

%I #19 Oct 15 2024 19:50:06

%S 2,7,17,33,57,93,149,241,401,693,1245,2313,4409,8557,16805,33249,

%T 66081,131685,262829,525049,1049417,2098077,4195317,8389713,16778417,

%U 33555733,67110269,134219241,268437081,536872653,1073743685,2147485633,4294969409,8589936837

%N a(n) = 2^n + 2*n^2 + 2*n + 1.

%C 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.

%H Marius A. Burtea, <a href="/A321123/b321123.txt">Table of n, a(n) for n = 0..201</a>

%H Ronald Cools, <a href="http://nines.cs.kuleuven.be/ecf/ndim_C.html"> Encyclopaedia of Cubature Formulas</a>

%H 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.

%H Alan C. Genz and Awais A. Malik, <a href="https://doi.org/10.1016/0771-050X(80)90039-X">Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region</a>, Journal of Computational and Applied Mathematics Vol. 6 (1980), 295-302.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).

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

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

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

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

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

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

%o (Magma) [2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // _Marius A. Burtea_, Dec 28 2018

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

%K nonn,easy

%O 0,1

%A _Franck Maminirina Ramaharo_, Dec 17 2018