A322593 a(n) = 2^n + 2*n^2 + 1.

2, 5, 13, 27, 49, 83, 137, 227, 385, 675, 1225, 2291, 4385, 8531, 16777, 33219, 66049, 131651, 262793, 525011, 1049377, 2098035, 4195273, 8389667, 16778369, 33555683, 67110217, 134219187, 268437025, 536872595, 1073743625, 2147485571, 4294969345, 8589936771

Offset: 0

Author

Franck Maminirina Ramaharo, Dec 18 2018

Keywords

• nonn
• easy

Comments

For n = 3..7, a(n) is the number of evaluating points on the n-dimensional sphere (also n-space with weight function exp(-r^2) or exp(-r)) in a degree 7 cubature rule.

References

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

Links

• Marius A. Burtea, Table of n, a(n) for n = 0..200
• Ronald Cools, Encyclopaedia of Cubature Formulas
• Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Cf. A000051, A000079, A001580, A001787, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A321123.

Programs

Magma

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

Mathematica

Table[2^n + 2*n^2 + 1, {n, 0, 50}]
LinearRecurrence[{5,-9,7,-2},{2,5,13,27},50] (* Harvey P. Dale, Mar 23 2021 *)

Maxima

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

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.

a(n) = a(n-1) + A100315(n-1), n >= 2.

G.f.: (2 - 5*x + 6*x^2 - 7*x^3)/((1 - 2*x)*(1 - x)^3)

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