0,3

Number of equilateral triangles in rhombic portion of side n+1 in hexagonal lattice.

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Sum of squared distances on n X n board between n queens each on her own row and column. - Zak Seidov (zakseidov(AT)yahoo.com), Sep 04 2002

For queens "each on her column and row" the sum of squared distances does not depend on configuration - while sum of distances does.

Number of cycles of length 3 in the bishop's graph associated with an (n+1) X (n+1) chessboard. [From Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009]

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 6).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

J. Propp, Updated article

J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

G.f.: 2x^2(1+x)/(1-x)^5.

a(n) = 2*A002415(n) = A047928(n-1)/6 = A083374(n-1)/3 = A006011(n)*2/3. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

a(2)=2 because on 2 X 2 board queens "each on her column and row" may take only two angular cells, then squared distance is 1^2+1^2=2. a(3)=12 because on 3 X 3 board queens "each on her column and row" make only two essentially distinct configurations: {1,2,3}, {1,3,2} and in both cases the sum of three squared distances is 12.

A008911 := n->n^2*(n^2-1)/6;

[seq(binomial(n+2, n+1)*binomial(n+3, n), n=-2..42)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 10 2006

a:=n->sum(sum(n^2/6, j=2..n), k=0..n): seq(a(n), n=0..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

a:=n->(sum((numbcomp(n, 4)), j=3..n)):seq(a(n), n=2..39); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]

a:=n->add(binomial(n, 2)+add(binomial(n, 2), j=1..n), j=1..n):seq(a(n)/3, n=0..35); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 27 2008]

a[m_] := m^2(m^2-1)/6

Table[Sum[Binomial[n, 3], {i, 2, n}], {n, 1, 41}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

(PARI) a(n)=n^2*(n^2-1)/6

Twice A002415.

Cf. A002415, A006011, A047928, A083374.

A002415, A083374 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 27 2008]

Sequence in context: A009632 A086602 A019006 this_sequence A005719 A118417 A143126

Adjacent sequences: A008908 A008909 A008910 this_sequence A008912 A008913 A008914

nonn

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy