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Minimal number of disjoint edge-paths into which the graph of the n-ary cube can be partitioned.
4

%I #15 Mar 27 2022 10:50:20

%S 1,1,4,1,16,1,64,1,256,1,1024,1,4096,1,16384,1,65536,1,262144,1,

%T 1048576,1,4194304,1,16777216,1,67108864,1,268435456,1,1073741824,1,

%U 4294967296,1,17179869184,1,68719476736,1,274877906944,1,1099511627776,1,4398046511104

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

%C 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).

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

%H Vincenzo Librandi, <a href="/A059222/b059222.txt">Table of n, a(n) for n = 1..400</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).

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

%F 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

%F E.g.f.: cosh(x)*(1 + sinh(x)) - 1. - _Stefano Spezia_, Mar 26 2022

%e 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.

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

%Y Cf. A057979.

%K nonn,easy

%O 1,3

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

%E More terms from _Harvey P. Dale_, Nov 02 2011