A102000 a(0),...,a(3) = 1, 2, 4, 8; thereafter a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3) + 8*a(n-4), n>3.

1, 2, 4, 8, 32, 80, 208, 560, 1552, 4144, 11152, 30128, 81424, 219440, 592016, 1597616, 4310800, 11629616, 31377808, 84661168, 228421648, 616292144, 1662802576, 4486362800, 12104509712, 32658782768, 88115674000, 237742180784

Offset: 0

Author

Gary W. Adamson, Dec 23 2004

Keywords

• nonn
• easy

Comments

Based on taking the n-th power of the matrix M = [1 1 1 1 / 2 0 0 0 / 0 2 0 0 / 0 0 2 0] that generates the D_4 lattice.

a(n)/a(n-1) tends to 2.698068913... an eigenvalue of M and a root of the characteristic polynomial x^4 - x^3 - 2*x^2 - 4*x - 8.

Links

• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition, 1999. (See Chap. 4.)
• Index entries for linear recurrences with constant coefficients, signature (1,2,4,8).

Programs

Mathematica

LinearRecurrence[{1, 2, 4, 8}, {1, 2, 4, 8}, 28] (* Hugo Pfoertner, Dec 11 2022 *)

Formula

G.f.: (-1-x+4*x^3)/(-1+x+2*x^2+4*x^3+8*x^4). [R. J. Mathar, Feb 13 2010]

Extensions

More terms from R. J. Mathar, Feb 13 2010

Edited by N. J. A. Sloane, Dec 11 2022