A052919 -id:A052919 - cp's OEIS frontend

A169724 (2*3^(n-1)+1)^2.

9, 49, 361, 3025, 26569, 237169, 2128681, 19140625, 172213129, 1549760689, 13947373801, 125524947025, 1129720271689, 10167469690609, 91507188951721, 823564585774225, 7412080927594249, 66708727315226929, 600378542737678441, 5403406875341014225

Offset: 1

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

A subsequence of the squares A000290.

Essentially equal to A052919(n)^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Alice V. Kleeva, Grid for this sequence
Alice V. Kleeva, Illustration of initial terms
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva
Robert Munafo, Sequence A169720, and two others by Alice V. Kleeva [Cached copy, in pdf format, included with permission]
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Cf. A169720-A169727, A100702.

I:=[9, 49, 361]; [n le 3 select I[n] else 13*Self(n-1) - 39*Self(n-2) + 27*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012

CoefficientList[Series[(-9 + 68 x - 75 x^2)/((x - 1) (3 x - 1) (9 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 03 2012 *)
LinearRecurrence[{13,-39,27},{9,49,361},20] (* Harvey P. Dale, Apr 23 2024 *)

a(n)= +13*a(n-1) -39*a(n-2) +27*a(n-3). G.f.: x*( -9+68*x-75*x^2 ) / ( (x-1)*(3*x-1)*(9*x-1) ). [R. J. Mathar, Apr 26 2010]

G.f. adapted to the offset by Vincenzo Librandi, Dec 03 2012

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.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

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

			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.

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

Cf. A000079, A052548, A008776.

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 *)

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.

cp's OEIS Frontend

A169724 (2*3^(n-1)+1)^2.

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