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A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

%I A384853 #12 Jul 21 2025 05:18:44
%S A384853 1,5,9,21,57,165,489,1461,4377,13125,39369,118101,354297,1062885,
%T A384853 3188649,9565941,28697817,86093445,258280329,774840981,2324522937,
%U A384853 6973568805,20920706409,62762119221,188286357657,564859072965,1694577218889,5083731656661
%N A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.
%C A384853 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).
%C A384853 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:
%C A384853     U             V         volume   squared diagonal length, ||g(n)||^2
%C A384853 (1, 0, 0)     (0, 1, 0)     A000079           A052548
%C A384853 (1, 0, 0)     (0, 1, 1)     A008776         3*A052919
%C A384853 (1, 0, 0)     (1, 0, 1)     A000351           A178676
%C A384853 (1, 0, 0)     (1, 1, 1)     A167747         5*A204061
%C A384853 (1, 0, 0)     (0, 2, 0)     A005054         4*A199215
%C A384853 (1, 0, 0)     (1, 2, 0)     A013731         8*A199552
%C A384853 (1, 0, 0)     (2, 1, 0)     A011557        10*A000533
%C A384853 (1, 0, 0)     (1, 1, 2)     A067403        18*A135423
%C A384853 (1, 0, 0)     (2, 1, 1)     A334603        11*A199750
%C A384853 (1, 0, 1)     (0, 1, 0)     A008776           this sequence
%C A384853 (1, 1, 0)     (0, 1, 1)     A081341         6*A199318
%C A384853 (1, 1, 0)     (1, 1, 1)     A270369         9*A199559
%C A384853 (1, 2, 3)     (3, 2, 1)   2*A009992   48 + 96*A009992
%H A384853 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A384853 a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
%F A384853 a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
%F A384853 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.
%e A384853 Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
%e A384853 (f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
%e A384853 (g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
%e A384853 (h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
%e A384853 The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
%e A384853 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.
%t A384853 f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
%t A384853 f[n_] := f[n] = h[n - 1];
%t A384853 g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
%t A384853 h[n_] := h[n] = Cross[f[n], g[n]];
%t A384853 v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
%t A384853 d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
%t A384853 Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
%t A384853 Table[v[n], {n, 1, 16}]  (* A008776 *)
%t A384853 u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
%t 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 *)
%Y A384853 Cf. A000079, A052548, A008776.
%K A384853 nonn
%O A384853 1,2
%A A384853 _Clark Kimberling_, Jul 02 2025