%I A006527 M3410
%S A006527 0,1,4,11,24,45,76,119,176,249,340,451,584,741,924,1135,1376,1649,
%T A006527 1956,2299,2680,3101,3564,4071,4624,5225,5876,6579,7336,8149,9020,
%U A006527 9951,10944,12001,13124,14315,15576,16909,18316,19799,21360,23001,24724
%N A006527 (n^3 + 2*n)/3.
%C A006527 Number of ways to color vertices of a triangle using <= n colors, allowing 
               only rotations.
%C A006527 Also: dot_product (1,2,...,n)*(2,3,...,n,1), n >= 0 (Clark Kimberling 
               ck6(AT)evansville.edu).
%C A006527 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
%C A006527 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
%C A006527 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
%C A006527 Starting with offset 1 = row sums of triangle A158822 and binomial transform 
               of (1, 3, 4, 2, 0, 0, 0,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 28 2009]
%D A006527 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006527 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.
%D A006527 M. Gardner, New Mathematical Diversions from Scientific American. Simon 
               and Schuster, NY, 1966, p. 246.
%D A006527 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (11).
%D A006527 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; 
               see p. 483.
%H A006527 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006527 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A006527 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006527 Th. Gruner, A. Kerber, R. Laue and M. Meringer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/
               meringer/pdf/MathCombChemSCCE.pdf">Mathematics for Combinatorial 
               Chemistry</a>
%F A006527 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
%F A006527 a(n)=n*A059100(n)/3. - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 06 
               2007
%F A006527 Starting (1, 4, 11, 24,...), = row sums of triangle A135184 - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007
%F A006527 a(n)= A054602(n)/3 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 
               20 2008
%p A006527 A006527:=z*(1+z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%p A006527 with(combinat):seq(lcm(fibonacci(4,n),fibonacci(2,n))/3,n=0..42); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008
%t A006527 Table[ (n^3 + 2*n)/3, {n, 0, 45} ]
%Y A006527 (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.
%Y A006527 Column 1 of triangle A094414. Row 6 of the array in A107735.
%Y A006527 Cf. A000292.
%Y A006527 Cf. A135184.
%Y A006527 A158822 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
%Y A006527 Sequence in context: A008250 A099074 A014818 this_sequence A057304 A001752 
               A160860
%Y A006527 Adjacent sequences: A006524 A006525 A006526 this_sequence A006528 A006529 
               A006530
%K A006527 nonn,easy,nice
%O A006527 0,3
%A A006527 N. J. A. Sloane (njas(AT)research.att.com).
%E A006527 More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), May 20 2006