A059222 Minimal number of disjoint edge-paths into which the graph of the n-ary cube can be partitioned.

1, 1, 4, 1, 16, 1, 64, 1, 256, 1, 1024, 1, 4096, 1, 16384, 1, 65536, 1, 262144, 1, 1048576, 1, 4194304, 1, 16777216, 1, 67108864, 1, 268435456, 1, 1073741824, 1, 4294967296, 1, 17179869184, 1, 68719476736, 1, 274877906944, 1, 1099511627776, 1, 4398046511104

Offset

1,3

Comments

The formula for this sequence is easily derived from a generalization of Euler's famous "Eulerian Path" theorem (see Theorem 11.2.4 in p. 419 of the reference).

References

R. A. Brualdi, Introductory Combinatorics, 3rd ed. Prentice-Hall, 1999.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Formula

a(n) = 1 if n is even and 2^(n-1) if n is odd.

G.f. -x*(-1-x+x^2+4*x^3) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x) ). - R. J. Mathar, Apr 25 2013

E.g.f.: cosh(x)*(1 + sinh(x)) - 1. - Stefano Spezia, Mar 26 2022

Example

a(5)=16 because 2^(5-1)=16. Consequently, the minimal number of disjoint edge-paths into which the 5-ary cube can be partitioned is 16.

Mathematica

Table[If[EvenQ[n], 1, 2^(n-1)], {n, 80}] (* or *) Riffle[2^(2Range[0, 50]), 1] (* Harvey P. Dale, Nov 02 2011 *)

Crossrefs

Cf. A057979.

Sequence in context: A123382 A197653 A146160 * A117292 A062780 A262616

Adjacent sequences: A059219 A059220 A059221 * A059223 A059224 A059225

Keyword

nonn,easy

Author

Felix Golderg (felixg(AT)tx.technion.ac.il), Jan 19 2001

Extensions

More terms from Harvey P. Dale, Nov 02 2011

Status

approved