%I A002492 M3562 N1444
%S A002492 0,4,20,56,120,220,364,560,816,1140,1540,2024,2600,3276,4060,4960,5984,
%T A002492 7140,8436,9880,11480,13244,15180,17296,19600,22100,24804,27720,30856,
%U A002492 34220,37820,41664,45760,50116,54740,59640,64824,70300,76076,82160
%N A002492 2n(n+1)(2n+1)/3.
%C A002492 Total number of possible bishop moves on an n X n chessboard, if the 
               bishop is placed anywhere. E.g. on a 3 X 3-Board: bishop has 8 X 
               2 moves and 1 X 4 moves, so a(3)=20. -Ulrich Schimke (ulrschimke(AT)aol.com)
%C A002492 Let M_n denotes the n X n matrix M_n(i,j)=(i+j)^2; then the characteristic 
               polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 
               14 2002
%C A002492 0,4,20,56,120 gives the number of electrons in closed shells in the double 
               shell periodic system of elements. This is a new interpretation of 
               the periodic system of the elements. The factor 4 in the formula 
               4*n(n+1)(2n+1)/6 plays a significant role, since it designates the 
               degeneracy of electronic states in this system. Closed shells with 
               more than 120 electrons are not expected to exist. - Karl-Dietrich 
               Neubert (kdn(AT)neubert.net)
%D A002492 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002492 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002492 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 A002492 A. O. Barut, Group Structure of the Periodic System, in Wybourne, Ed., 
               The Structure of Matter, University of Canterbury Press, Christchurch, 
               1972, p. 126.
%D A002492 Edward G. Mazur, Graphic Representation of the Periodic System during 
               One Hundred Years, University of Alabama Press, Alabama, 1974
%D A002492 D. Neubert, Double Shell Structure of the Periodic System of the Elements, 
               Z. Naturforschung, 25A (1970) p. 210.
%H A002492 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A002492 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A002492 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 A002492 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 A002492 Karl-Dietrich Neubert, <a href="http://www.neubert.net/PSEMetal.html">
               Title?</a>
%H A002492 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A002492 G.f.: 4x(1+x)/(1-x)^4. a(-1-n)=-a(n).
%F A002492 Partial sums of A016742. a(n)=C(2*(n+1), 3). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 19 2004
%F A002492 Equals C(2+2xn,3) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 
               12 2006
%p A002492 A002492:=4*(1+z)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
%t A002492 s = 0; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 2, 80, 2}]; lst 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
%o A002492 (PARI) a(n)=2*n*(n+1)*(2*n+1)/3
%Y A002492 a(n)= (-1)^(n+1)*A053120(2*n+1, 3) (fourth unsigned column of Chebyshev 
               T-triangle, zeros omitted).
%Y A002492 a(n) = 4*A000330(n) = A000292(2n+1). Cf. A006331.
%Y A002492 Cf. A033586, A035006, A035008.
%Y A002492 Sequence in context: A023667 A035007 A047810 this_sequence A127920 A060122 
               A066970
%Y A002492 Adjacent sequences: A002489 A002490 A002491 this_sequence A002493 A002494 
               A002495
%K A002492 nonn,easy,nice
%O A002492 0,2
%A A002492 N. J. A. Sloane (njas(AT)research.att.com).