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 A016742 Even squares: a(n) = (2*n)^2. 135

%I

%S 0,4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156,

%T 1296,1444,1600,1764,1936,2116,2304,2500,2704,2916,3136,3364,3600,

%U 3844,4096,4356,4624,4900,5184,5476,5776,6084,6400,6724,7056,7396,7744,8100,8464

%N Even squares: a(n) = (2*n)^2.

%C 4 times the squares.

%C Number of edges in the complete bipartite graph of order 5n, K_{n,4n} - _Roberto E. Martinez II_, Jan 07 2002

%C It is conjectured (I think) that a regular Hadamard matrix of order n exists iff n is an even square (cf. Seberry and Yamada, Th. 10.11). A Hadamard matrix is regular if the sum of the entries in each row is the same. - _N. J. A. Sloane_, Nov 13 2008

%C Sequence arises from reading the line from 0, in the direction 0, 16, ... and the line from 4, in the direction 4, 36, ... in the square spiral whose vertices are the squares A000290. - _Omar E. Pol_, May 24 2008

%C The entries from a(1) on can be interpreted as pair sums of (2, 2), (8, 8), (18, 18), (32, 32) etc. that arise from a re-arrangement of the subshell orbitals in the periodic table of elements. 8 becomes the maximum number of electrons in the (2s,2p) or (3s,3p) orbitals, 18 the maximum number of electrons in (4s,3d,4p) or (5s,3d,5p) shells, for example. - _Julio Antonio Gutiérrez Samanez_, Jul 20 2008

%C The first two terms of the sequence (n=1, 2) give the numbers of chemical elements using only n types of atomic orbitals, i.e., there are a(1)=4 elements (H,He,Li,Be) where electrons reside only on s-orbitals, there are a(2)=16 elements (B,C,N,O,F,Ne,Na,Mg,Al,Si,P,S,Cl,Ar,K,Ca) where electrons reside only on s- and p-orbitals. However, after that, there is 37 (which is one more than a(3)=36) elements (from Sc, Scandium, atomic number 21 to La, Lanthanum, atomic number 57) where electrons reside only on s-, p- and d-orbitals. This is because Lanthanum (with the electron configuration [Xe]5d^1 6s^2) is an exception to the Aufbau principle, which would predict that its electron configuration is [Xe]4f^1 6s^2. - _Antti Karttunen_, Aug 14 2008.

%C Number of cycles of length 3 in the king's graph associated with an (n+1) X (n+1) chessboard. - Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009

%C The sum to infinity of the reciprocals of the members of this sequence converges to (1/4)*Pi^2/6 = Pi^2/24. - _Ant King_, Nov 04 2009

%C a(n+1) is the molecular topological index of the n-star graph S_n. - _Eric W. Weisstein_, Jul 11 2011

%C a(n) is the sum of two consecutives odd numbers 2*n^2-1 and 2*n^2+1 and the difference of two squares (n^2+1)^2 - (n^2-1)^2. - _Pierre CAMI_, Jan 02 2012

%C For n > 3, a(n) is the area of the irregular quadrilateral created by the points ((n-4)*(n-3)/2,(n-3)*(n-2)/2), ((n-2)*(n-1)/2,(n-1)*n/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+3)*(n+4)/2,(n+2)*(n+3)/2). - _J. M. Bergot_, May 27 2014

%C a(n) is the least number k = x + y, with x>0 and y>0, such that there are n different pairs (x,y) for which x*y/k is an integer. - _Paolo P. Lava_, Jan 26 2018

%C Number of terms less than 10^k: 1, 2, 5, 16, 50, 159, 500, 1582, 5000, 15812, 50000, 158114, 500000, ... - _Muniru A Asiru_, Jan 28 2018

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

%D Seberry, Jennifer and Yamada, Mieko; Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.

%D W. D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Combinatorics: Room squares, sum-free sets, Hadamard matrices, Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin-New York, 1972. iv+508 pp.

%H Vincenzo Librandi, <a href="/A016742/b016742.txt">Table of n, a(n) for n = 0..900</a>

%H R. P. Boas and N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>.

%H Various, <a href="http://www.physicsforums.com/showthread.php?t=240722">Electron Configuration</a> (Discussion in Physics Forums).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MolecularTopologicalIndex.html">Molecular Topological Index</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Aufbau_principle">Aufbau principle</a>.

%H <a href="/index/Ha#Hadamard">Index entries for sequences related to Hadamard matrices</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F O.g.f.: 4*x*(1+x)/(1-x)^3. - _R. J. Mathar_, Jul 28 2008

%F a(n) = A000290(n)*4 = A001105(n)*2. - _Omar E. Pol_, May 21 2008

%F a(n) = A155955(n,2) for n > 1. - _Reinhard Zumkeller_, Jan 31 2009

%F a(n) = a(n-1) + 8*n - 4 (with a(0)=0). - _Vincenzo Librandi_, Nov 19 2010

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 4, a(2) = 16. - _Philippe Deléham_, Mar 26 2013

%F a(n) = A118729(8n+3). - _Philippe Deléham_, Mar 26 2013

%F Pi = 2*Product_{n>=1} (1 + 1/(a(n)-1)). - _Adriano Caroli_, Aug 04 2013

%F Pi = Sum_{n>=0} 8/(a(2n+1)-1). - _Adriano Caroli_, Aug 06 2013

%F E.g.f.: exp(x)*(4x^2 + 4x). - _Geoffrey Critzer_, Oct 07 2013

%F a(n) = A000384(n) + A014105(n). - _Bruce J. Nicholson_, Nov 11 2017

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 (A245058). - _Amiram Eldar_, Oct 10 2020

%F From _Amiram Eldar_, Jan 25 2021: (Start)

%F Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/2)/(Pi/2) (A308716).

%F Product_{n>=1} (1 - 1/a(n)) = sin(Pi/2)/(Pi/2) = 2/Pi (A060294). (End)

%p seq((2*n)^2, n=0..100); # _Muniru A Asiru_, Jan 28 2018

%t Table[(2n)^2, {n, 0, 46}] (* _Alonso del Arte_, Apr 26 2011 *)

%o (MAGMA) [(2*n)^2: n in [0..50]]; // _Vincenzo Librandi_, Apr 26 2011

%o (Maxima) makelist((2*n)^2,n,0,20); /* _Martin Ettl_, Jan 22 2013 */

%o (Haskell)

%o a016742 = (* 4) . (^ 2)

%o a016742_list = 0 : map (subtract 4) (zipWith (+) a016742_list [8, 16 ..])

%o -- _Reinhard Zumkeller_, Jun 28 2015, Apr 20 2015

%o (PARI) a(n)=4*n^2 \\ _Charles R Greathouse IV_, Jul 28 2015

%o (GAP) List([0..100], n -> (2*n)^2); # _Muniru A Asiru_, Jan 28 2018

%Y Cf. A000290, A001105, A001539, A016754, A016802, A016814, A016826, A016838, A007742, A033991, A245058.

%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.

%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

%Y Cf. sequences listed in A254963.

%Y Other n X n king graph cycle counts: A288918 (4-cycles), A288919 (5-cycles), A288920 (6-cycles).

%Y Cf. A000384, A014105.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from Sabir Abdus-Samee (sabdulsamee(AT)prepaidlegal.com), Mar 13 2006

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Last modified April 10 07:36 EDT 2021. Contains 342843 sequences. (Running on oeis4.)