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

3,1

Links

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Vaclav Kotesovec, Recurrence (of order 11)

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

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 *)

Crossrefs

Column k=3 of A255982.

Sequence in context: A321045 A004416 A125487 * A212473 A127544 A133927

Adjacent sequences: A258414 A258415 A258416 * A258418 A258419 A258420

Keyword

nonn

Author

Alois P. Heinz, May 29 2015

Status

approved