0,2

Number of different directions along lines and hyper-diagonals in an n-dimensional cubic lattice for the attacking queens problem (A036464 in n=2, A068940 in n=3 and A068941 in n=4). The n-dimensional direction vectors have the a(n)+1 Cartesian coordinates (i,j,k,l,..) where i,j,k,l..=-1,0 or +1, excluding the zero-vector i=j=k=l=..=0. The corresponding hyper-line count is A003462. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 01 2006

Numbers n for which the expression 3^n/(n+1) is an integer. - Paolo P. Lava (ppl(AT)spl.at), May 29 2006

A128760(a(n)) > 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 25 2007

Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition sum(|n_k|,k=1..m) <= n. See the K. A. Meissner example in arXiv:gr-qc/0407052v1, p. 6 (with a typo: it should be 3^([2a]-1)-1). W. Lang, Jan 21 2008.

a(n)=2*A003462(n) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 01 2006

G.f.: 2*x/(-1+x)/(-1+3*x) = 1/(-1+x)-1/(-1+3*x). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 19 2007

a(n)=sum(sum(binomial(k-1,m-1)*2^m,m=1..k),k=1..n), n>=1. a(0)=0. From the sequence combinatorics mentioned above. Twice partial sums of powers of 3.

Ternary......decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007):

0...............0

2...............2

22..............8

222............26

2222...........80

22222.........242

222222........728

2222222......2186

22222222.....6560

222222222...19682

2222222222..59048

etc...........etc.

Sequence combinatorics: n=3: With length m=1: [1],[2],[3] each with 2 signs, with m=2: [1,1], [1,2], [2,1], each 2^2=4 times from choosing signs; m=3: [1,1,1] coming in 2^3 signed versions: 3*2+3*4+1*8= 26 = a(3). The order is important, hence the M_0 multinomials A048996 enter as factors.

with(finance):seq(add(futurevalue( 2, 2, k), k=0..n), n=-1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

Cf. triangle A013609.

Sequence in context: A136594 A097040 A124721 this_sequence A103453 A126966 A002930

Adjacent sequences: A024020 A024021 A024022 this_sequence A024024 A024025 A024026

nonn

njas