A258417 Number of partitions of the 3-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

30, 486, 5880, 64464, 679195, 7043814, 72707844, 751082244, 7785793080, 81092511276, 849060054420, 8937364804760, 94564644817767, 1005496779910572, 10740560345206680, 115218669255806304, 1240869923563291014, 13412271463669969704, 145454088924589697192

Offset: 3

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000
• Vaclav Kotesovec, Recurrence (of order 11)

Crossrefs

Column k=3 of A255982.

Programs

Maple

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(n, 3):
seq(a(n), n=3..25);

Mathematica

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 3];
a /@ Range[3, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 11.6335027253872064795086728699206569842475549795979388187955249065144... is the root of the equation 16777216 - 150994944*d + 1716387840*d^3 + 2063339520*d^4 - 6994944*d^5 - 21019200*d^6 + 454313*d^7 = 0 and c = 0.6170954330535517584816422123448632671500498041324155957832713069267... . - Vaclav Kotesovec, Feb 20 2016