1,2

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).

The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:

U V volume squared diagonal length, ||g(n)||^2

(1, 0, 0) (0, 1, 0) A000079 A052548

(1, 0, 0) (0, 1, 1) A008776 3*A052919

(1, 0, 0) (1, 0, 1) A000351 A178676

(1, 0, 0) (1, 1, 1) A167747 5*A204061

(1, 0, 0) (0, 2, 0) A005054 4*A199215

(1, 0, 0) (1, 2, 0) A013731 8*A199552

(1, 0, 0) (2, 1, 0) A011557 10*A000533

(1, 0, 0) (1, 1, 2) A067403 18*A135423

(1, 0, 0) (2, 1, 1) A334603 11*A199750

(1, 0, 1) (0, 1, 0) A008776 this sequence

(1, 1, 0) (0, 1, 1) A081341 6*A199318

(1, 1, 0) (1, 1, 1) A270369 9*A199559

(1, 2, 3) (3, 2, 1) 2*A009992 48 + 96*A009992

Index entries for linear recurrences with constant coefficients, signature (4,-3).

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.

a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.

In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.

Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by

(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)

(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)

(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)

The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).

The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];

f[n_] := f[n] = h[n - 1];

g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];

h[n_] := h[n] = Cross[f[n], g[n]];

v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)

d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)

Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]] (* edge vectors of P(n) *)

Table[v[n], {n, 1, 16}] (* A008776 *)

u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)

Join[{1}, Table[1+2*(3^(n-1)+1), {n, 40}]] (* or *) LinearRecurrence[{4, -3}, {1, 5, 9}, 50] (* Harvey P. Dale, Jul 20 2025 *)

Cf. A000079, A052548, A008776.

Sequence in context: A097074 A050559 A082676 * A146932 A146476 A110200

Adjacent sequences: A384850 A384851 A384852 * A384854 A384855 A384856

Clark Kimberling, Jul 02 2025

approved