Search: id:A006331
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%I A006331 M1963
%S A006331 0,2,10,28,60,110,182,280,408,570,770,1012,1300,1638,2030,2480,2992,
%T A006331 3570,4218,4940,5740,6622,7590,8648,9800,11050,12402,13860,15428,17110,
%U A006331 18910,20832,22880,25058,27370,29820,32412,35150,38038,41080,44280
%N A006331 n*(n+1)*(2n+1)/3.
%C A006331 Triangles in rhombic matchstick arrangement of side n.
%C A006331 Maximum accumulated number of electrons at energy level n - Scott A. 
               Brown (scottbrown(AT)neo.rr.com), Feb 28 2000.
%C A006331 Let M_n denotes the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic 
               polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 
               14 2002
%C A006331 Convolution of odds (A005408) and evens (A005843) - Graeme McRae (g_m(AT)mcraefamily.com), 
               Jun 06 2006
%D A006331 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006331 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 A006331 G. Kreweras, Sur une classe de problemes de denombrement lies au treillis 
               des partitions des entiers, Cahiers du Bureau Universitaire de Recherche 
               Op\'{e}rationnelle, Institut de Statistique, Universit\'{e} de Paris, 
               6 (1965), circa p. 82.
%H A006331 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 A006331 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 A006331 Author?, <a href="http://users.senet.com.au/~rowanb/chem/atstruct.html">
               Basic atomic information</a>
%F A006331 G.f.: x*(2+2*x)/(1-x)^4. a(n)=2*C(n+1, 3)+2*C(n+2, 3).
%F A006331 From the formula for the sum of squares of positive integers 1^2+2^2+3^2+...+n^2 
               = n(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum(2*k^2,
               k=0..n) = n(n+1)(2*n+1)/3, which is an alternative formula for this 
               sequence. - Mike Warburton (mikewarb(AT)gmail.com), Sep 08 2007
%p A006331 A006331:=2*(1+z)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
%t A006331 f[n_]:=4*n+2; s1=s2=0;lst={};Do[a=f[n];s1+=a;s2+=s1;AppendTo[lst,s2],
               {n,0,6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 
               25 2009]
%t A006331 s = 0; lst = {s}; Do[s += 2*n^2; AppendTo[lst, s], {n, 1, 40, 1}]; lst 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 15 2009]
%o A006331 (PARI) a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
%Y A006331 2*A000330. Cf. A002492. A row of A132339.
%Y A006331 a(n)=Sum{T(i, n-i): i=0, 1, ..., n}, array T as in A048147.
%Y A006331 Sequence in context: A060515 A109723 A053594 this_sequence A104657 A000900 
               A124023
%Y A006331 Adjacent sequences: A006328 A006329 A006330 this_sequence A006332 A006333 
               A006334
%K A006331 nonn,easy,nice
%O A006331 0,2
%A A006331 N. J. A. Sloane (njas(AT)research.att.com).

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