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A002492 Sum of the first n even squares: 2*n*(n+1)*(2*n+1)/3.
(Formerly M3562 N1444)
26
0, 4, 20, 56, 120, 220, 364, 560, 816, 1140, 1540, 2024, 2600, 3276, 4060, 4960, 5984, 7140, 8436, 9880, 11480, 13244, 15180, 17296, 19600, 22100, 24804, 27720, 30856, 34220, 37820, 41664, 45760, 50116, 54740, 59640, 64824, 70300, 76076, 82160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Total number of possible bishop moves on an n+1 X n+1 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(2)=20. - Ulrich Schimke (ulrschimke(AT)aol.com)

Let M_n denote 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

Partial sums of A016742. - Lekraj Beedassy, Jun 19 2004

Obviously A035005(n+1) = a(n) + A035006(n+1) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010

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)

Inverse binomial transform of A240434. - Wesley Ivan Hurt, Apr 13 2014

For n>2, a(n) = Sum_{k=0..3} C(n-2+k,n-2)*C(n+3-k,n). - J. M. Bergot, Jun 14 2014

Atomic number of alkaline-earth metals of period 2n. - Natan Arie' Consigli, Jul 03 2016

REFERENCES

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.

W. Permans and J. Kemperman, "Nummeringspribleem van S. Dockx, Mathematisch Centrum. Amsterdam," Rapport ZW; 1949-005, 4 leaves, 19.8 X 34 cm.

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

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

D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226; http://dx.doi.org/10.12988/ams.2014.4140.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

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

Karl-Dietrich Neubert, Double-Shell PSE: Metals - Nonmetals

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: 4*x*(1+x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation

a(-1-n) = -a(n).

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

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

a(n) = binomial(2*n+2, 3). - Lekraj Beedassy, Jun 19 2004

a(n) - a(n-1) = 4*n^2. - Joerg Arndt, Jun 16 2011

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Aug 15 2012

a(n) = 2*A006331(n). - R. J. Mathar, May 28 2016

From Natan Arie' Consigli Jul 03 2016: (Start)

a(n) = A166464(n) - 1.

a(n) = A168380(2*n). (End)

a(n) = Sum_{i=0..n} A005408(i)*A005408(i-1)+1 with A005408(-1):=-1. - Bruno Berselli, Jan 09 2017

MAPLE

A002492:=n->2*n*(n+1)*(2*n+1)/3; seq(A002492(n), n=0..50); # Wesley Ivan Hurt, Apr 04 2014

MATHEMATICA

Table[2n(n+1)(2n+1)/3, {n, 0, 40}] (* or *) Binomial[2*Range[0, 40]+2, 3] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 4, 20, 56}, 40] (* Harvey P. Dale, Aug 15 2012 *)

PROG

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

(MAGMA) [2*n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

CROSSREFS

Cf. A000292, A000330, A005408, A006331, A053120.

Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A049450 (Pawn).

Sequence in context: A213485 A035007 A047810 * A127920 A060122 A066970

Adjacent sequences:  A002489 A002490 A002491 * A002493 A002494 A002495

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Comment added, minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010

Title modified by Charles R Greathouse IV on the suggestion of J. M. Bergot, Apr 05 2014

STATUS

approved

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Last modified July 21 20:57 EDT 2017. Contains 289648 sequences.