A140320 a(n) = A137576((3^n-1)/2).

1, 3, 13, 55, 217, 811, 2917, 10207, 34993, 118099, 393661, 1299079, 4251529, 13817467, 44641045, 143489071, 459165025, 1463588515, 4649045869, 14721978583, 46490458681, 146444944843, 460255540933, 1443528742015, 4518872583697, 14121476824051, 44059007691037, 137260754729767

Offset: 0

Author

Vladimir Shevelev, May 26 2008

Keywords

• nonn

Comments

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

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

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]

Links

• Wikipedia, Hypercube

Crossrefs

Cf. A037576, A002326, A006694, A025192.

Programs

PARI

a137576(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1;
a(n) = a137576((3^n-1)/2); \\ Michel Marcus, Dec 18 2018

Formula

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]

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

Extensions

More terms from Michel Marcus, Dec 18 2018