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%I M3562 N1444 #176 Sep 01 2023 04:41:18
%S 0,4,20,56,120,220,364,560,816,1140,1540,2024,2600,3276,4060,4960,
%T 5984,7140,8436,9880,11480,13244,15180,17296,19600,22100,24804,27720,
%U 30856,34220,37820,41664,45760,50116,54740,59640,64824,70300,76076,82160
%N Sum of the first n even squares: 2*n*(n+1)*(2*n+1)/3.
%C 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)
%C 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
%C Partial sums of A016742. - _Lekraj Beedassy_, Jun 19 2004
%C 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)
%C Inverse binomial transform of A240434. - _Wesley Ivan Hurt_, Apr 13 2014
%C Atomic number of alkaline-earth metals of period 2n. - _Natan Arie Consigli_, Jul 03 2016
%C 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
%C 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
%D 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 Edward G. Mazur, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press, Alabama, 1974.
%D W. Permans and J. Kemperman, "Nummeringspribleem van S. Dockx, Mathematisch Centrum. Amsterdam," Rapport ZW; 1949-005, 4 leaves, 19.8 X 34 cm.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002492/b002492.txt">Table of n, a(n) for n = 0..1000</a>
%H Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, <a href="https://arxiv.org/abs/2307.13749">Semi-simplicial combinatorics of cyclinders and subdivisions</a>, arXiv:2307.13749 [math.CO], 2023. See p. 32.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014), Article #14.3.5.
%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H D. Neubert, <a href="http://www.neubert.net/PseNaturforsch/page110.pdf">Double Shell Structure of the Periodic System of the Elements</a>, Z. Naturforschung, 25A (1970), p. 210.
%H Karl-Dietrich Neubert, <a href="http://www.neubert.net/PSEMetal.html">Double-Shell PSE: Metals - Nonmetals</a>.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H D. Suprijanto and Rusliansyah, <a href="http://dx.doi.org/10.12988/ams.2014.4140">Observation on Sums of Powers of Integers Divisible by Four</a>, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2219-2226.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: 4*x*(1+x)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation
%F a(-1-n) = -a(n).
%F a(n) = 4*A000330(n) = 2*A006331(n) = A000292(2*n).
%F a(n) = (-1)^(n+1)*A053120(2*n+1,3) (fourth unsigned column of Chebyshev T-triangle, zeros omitted).
%F a(n) = binomial(2*n+2, 3). - _Lekraj Beedassy_, Jun 19 2004
%F A035005(n+1) = a(n) + A035006(n+1) since Queen = Bishop + Rook. - _Johannes W. Meijer_, Feb 04 2010
%F a(n) - a(n-1) = 4*n^2. - _Joerg Arndt_, Jun 16 2011
%F 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
%F 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
%F a(n) = 2*A006331(n). - _R. J. Mathar_, May 28 2016
%F From _Natan Arie Consigli_ Jul 03 2016: (Start)
%F a(n) = A166464(n) - 1.
%F a(n) = A168380(2*n). (End)
%F a(n) = Sum_{i=0..n} A005408(i)*A005408(i-1)+1 with A005408(-1):=-1. - _Bruno Berselli_, Jan 09 2017
%F a(n) = A002412(n) + A016061(n). - _Bruce J. Nicholson_, Nov 12 2017
%F From _Amiram Eldar_, Jan 04 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 9/2 - 6*log(2).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/2 - 9/2. (End)
%F 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
%F E.g.f.: 2*exp(x)*x*(6 + 9*x + 2*x^2)/3. - _Stefano Spezia_, Jul 31 2022
%p A002492:=n->2*n*(n+1)*(2*n+1)/3; seq(A002492(n), n=0..50); # _Wesley Ivan Hurt_, Apr 04 2014
%t 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 *)
%t Accumulate[(2*Range[0,40])^2] (* _Harvey P. Dale_, Jun 04 2019 *)
%o (PARI) a(n)=2*n*(n+1)*(2*n+1)/3
%o (Magma) [2*n*(n+1)*(2*n+1)/3: n in [0..40]]; // _Vincenzo Librandi_, Jun 16 2011
%Y Cf. A000292, A000330, A005408, A006331, A053120, A081277.
%Y Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A049450 (Pawn).
%Y Cf. A002412, A016061.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Minor errors corrected and edited by _Johannes W. Meijer_, Feb 04 2010
%E Title modified by _Charles R Greathouse IV_ at the suggestion of _J. M. Bergot_, Apr 05 2014
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