0,3

Number of ways to color vertices of a triangle using <= n colors, allowing only rotations.

Also: dot_product (1,2,...,n)*(2,3,...,n,1), n >= 0 (Clark Kimberling ck6(AT)evansville.edu).

Define a(n) by a(1) = 1 and a(n) = 1*2 + 2*3 + 3*4 +. . .+ (n-1)*n + n*1, the sum of the cyclic product of terms taken two at a time, final term being n*1; this gives same sequence. Example: a(3) = 1*2 + 2*3 + 3*1 = 11. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 02 2001

Start from triacid and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - rgwv, Aug 02 2002

a(n) = A000292(n-1) + A000292(n-3) - Sum of two tetrahedral numbers (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. - Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Th. Gruner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry

a(n)=2C(n+1, 3)+C(n, 1). G.f.: x(1+x^2)/(1-x)^4 - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003

a(n)=n*A059100(n)/3. - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 06 2007

Starting (1, 4, 11, 24,...), = row sums of triangle A135184 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007

a(n)= A054602(n)/3 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

A006527:=z*(1+z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

with(combinat):seq(lcm(fibonacci(4, n), fibonacci(2, n))/3, n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

Table[ (n^3 + 2*n)/3, {n, 0, 45} ]

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Column 1 of triangle A094414. Row 6 of the array in A107735.

Cf. A000292.

Cf. A135184.

Sequence in context: A008250 A099074 A014818 this_sequence A057304 A001752 A007678

Adjacent sequences: A006524 A006525 A006526 this_sequence A006528 A006529 A006530

nonn,easy,nice

njas

More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006