A002492 Sum of the first n even squares: 2*n*(n+1)*(2*n+1)/3.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

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

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

a(n) are the negative cubic coefficients in the expansion of sin(kx) into powers of sin(x) for the odd k: sin(kx) = k sin(x) - c(k) sin^3(x) + O(sin^5(x)); a(n) = c(2n+1) = A000292(2n). - Mathias Zechmeister, Jul 24 2022

Also the number of distinct series-parallel networks under series-parallel reduction on three unlabeled edges of n element kinds. - Michael R. Hayashi, Aug 02 2023

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.
• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 32.
• Milan Janjic, Two Enumerative Functions
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article #14.3.5.
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• 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, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2219-2226.
• 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).

Crossrefs

Cf. A000292, A000330, A005408, A006331, A053120, A081277.

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

Cf. A002412, A016061.

Programs

Magma

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

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 *)
Accumulate[(2*Range[0,40])^2] (* Harvey P. Dale, Jun 04 2019 *)

PARI

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

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

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

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

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

a(n) = A002412(n) + A016061(n). - Bruce J. Nicholson, Nov 12 2017

From Amiram Eldar, Jan 04 2022: (Start)

Sum_{n>=1} 1/a(n) = 9/2 - 6*log(2).

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/2 - 9/2. (End)

a(n) = A081277(3, n-1) = (1+2*n)*binomial(n+1, n-2)*2^2/(n-1) for n > 0. - Mathias Zechmeister, Jul 26 2022

E.g.f.: 2*exp(x)*x*(6 + 9*x + 2*x^2)/3. - Stefano Spezia, Jul 31 2022

Extensions

Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010

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