0,2

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)

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

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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

A. O. Barut, Group Structure of the Periodic System, in Wybourne, Ed., The Structure of Matter, University of Canterbury Press, Christchurch, 1972, p. 126.

Edward G. Mazur, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press, Alabama, 1974

D. Neubert, Double Shell Structure of the Periodic System of the Elements, Z. Naturforschung, 25A (1970) p. 210.

Index entries for two-way infinite sequences

Milan Janjic, Two Enumerative Functions

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.

Karl-Dietrich Neubert, Title?

Index entries for sequences related to Chebyshev polynomials.

G.f.: 4x(1+x)/(1-x)^4. a(-1-n)=-a(n).

Partial sums of A016742. a(n)=C(2*(n+1), 3). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 19 2004

Equals C(2+2xn,3) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2006

A002492:=4*(1+z)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

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]

(PARI) a(n)=2*n*(n+1)*(2*n+1)/3

a(n)= (-1)^(n+1)*A053120(2*n+1, 3) (fourth unsigned column of Chebyshev T-triangle, zeros omitted).

a(n) = 4*A000330(n) = A000292(2n+1). Cf. A006331.

Cf. A033586, A035006, A035008.

Sequence in context: A023667 A035007 A047810 this_sequence A127920 A060122 A066970

Adjacent sequences: A002489 A002490 A002491 this_sequence A002493 A002494 A002495

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).