A140320 - OEIS

%I #12 Dec 18 2018 11:28:41

%S 1,3,13,55,217,811,2917,10207,34993,118099,393661,1299079,4251529,

%T 13817467,44641045,143489071,459165025,1463588515,4649045869,

%U 14721978583,46490458681,146444944843,460255540933,1443528742015,4518872583697,14121476824051,44059007691037,137260754729767

%N a(n) = A137576((3^n-1)/2).

%C Conjecture. a(n) = 2n*3^(n-1)+1.

%C If conjecture is true then limsup(A137576(n)/n)=infinity while liminf(A137576(n)/n)=2 with a realization on primes.

%C a(n) is also the number of edges in the graph generated from the n-dimensional hypercube (plus 1) in the following manner: connect all (d + 1)-dimensional faces to the d faces that are incident. Each d-dimensional face should be incident on (n - d) (d + 1)-dimensional faces. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypercube">Hypercube</a>

%F Sum_{m = 0}^{n} 2^(n - m) * binomial(n,m) is the number of m-dimensional faces in the n-dimensional hypercube. Consequently, Sum_{m = 0..n} (n - m) * 2^(n - m) * binomial(n,m) gives the number of incidence edges, which yields said sequence minus 1. The recurrence relation is: a(n) = 3 * a(n - 1) + 2 * 3^(n - 1) - 2. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

%F Empirical G.f.: (1-4*x+7*x^2)/(1-7*x+15*x^2-9*x^3). [_Colin Barker_, Jan 09 2012]

%o (PARI) a137576(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1;

%o a(n) = a137576((3^n-1)/2); \\ _Michel Marcus_, Dec 18 2018

%Y Cf. A037576, A002326, A006694, A025192.

%K nonn

%O 0,2

%A _Vladimir Shevelev_, May 26 2008

%E More terms from _Michel Marcus_, Dec 18 2018