A398262 - OEIS

A398262

Number of indecomposable closed walks of length 2n along the edges of the 4-cube (hypercube Q_4, the tesseract) based at a vertex.

1, 4, 24, 288, 4032, 57600, 824832, 11814912, 169242624, 2424324096, 34727362560, 497454022656, 7125807661056, 102074026033152, 1462165032665088, 20944863897845760, 300025861581570048, 4297737051756822528, 61563172150151479296, 881865065160259928064, 12632324910960523345920

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

An indecomposable closed walk returns to its starting vertex exactly once, on the final step.

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (16,-24).

FORMULA

G.f.: 2 - 1/f(x), where f(x) is the g.f. for A092812 (closed walks on the 4-cube).

G.f.: (1 - 12*x - 16*x^2)/(1 - 16*x + 24*x^2).

a(n) = 16*a(n-1) - 24*a(n-2) for n > 2; a(0) = 1, a(1) = 4, a(2) = 24.

a(n) = A092812(n) - Sum_{k=1..n-1} a(k)*A092812(n-k) for n > 0 (first-return recurrence).

Characteristic roots of the recurrence are 8 +- 2*sqrt(10); hence a(n) grows like (8 + 2*sqrt(10))^n = 14.32455...^n.

E.g.f.: (exp(8*x)*(25*cosh(2*sqrt(10)*x) - 7*sqrt(10)*sinh(2*sqrt(10)*x)) - 10)/15. - Stefano Spezia, Jul 24 2026

EXAMPLE

a(1) = 4: from a fixed vertex of the tesseract, each of the 4 incident edges gives one length-2 out-and-back closed walk that first returns at step 2.

MATHEMATICA

nn = 40; list = Range[0, nn]! CoefficientList[Series[Cosh[x]^4, {x, 0, nn}], x];

a = Sum[list[[i]] x^(i - 1), {i, 1, nn + 1}];

Select[CoefficientList[Series[2 - 1/a, {x, 0, nn}], x], # > 0 &]

(* Alternative: *)

Join[{1}, LinearRecurrence[{16, -24}, {4, 24}, 30]]

PROG

(PARI) Vec((1 - 12*x - 16*x^2)/(1 - 16*x + 24*x^2) + O(x^25))

(Python)

a=[1, 4, 24]

for n in range(3, 25): a.append(16*a[-1]-24*a[-2])

print(a)