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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. 2
30, 486, 5880, 64464, 679195, 7043814, 72707844, 751082244, 7785793080, 81092511276, 849060054420, 8937364804760, 94564644817767, 1005496779910572, 10740560345206680, 115218669255806304, 1240869923563291014, 13412271463669969704, 145454088924589697192 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 20 22:00 EDT 2021. Contains 348119 sequences. (Running on oeis4.)