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A016742 Even squares: a(n) = (2n)^2. 106
0, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

4 times the squares.

Number of edges in the complete bipartite graph of order 5n, K_{n,4n} - Roberto E. Martinez II, Jan 07 2002

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

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

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

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.

a(n) = A155955(n,2) for n>1. - Reinhard Zumkeller, Jan 31 2009

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

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

a(n+1) is the molecular topological index of the n-star graph S_n. - Eric W. Weisstein, Jul 11 2011

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

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

REFERENCES

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

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.

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..900

R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Various, Electron Configuration (Discussion in Physics Forums)

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, King Graph

Eric Weisstein's World of Mathematics, Molecular Topological Index

Wikipedia, Aufbau principle

Index entries for sequences related to Hadamard matrices

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

O.g.f.: 4*x*(1+x)/(1-x)^3. - R. J. Mathar, Jul 28 2008

a(n) = A000290(n)*4 = A001105(n)*2. - Omar E. Pol, May 21 2008

a(n) = a(n-1) + 8*n - 4 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010

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

a(n) = A118729(8n+3). - Philippe Deléham, Mar 26 2013

Pi = 2*prod(n>=1, 1+1/(a(n)-1)) ). - Adriano Caroli, Aug 04 2013

Pi = sum_{n>=0} 8/(a(2n+1)-1). - Adriano Caroli, Aug 06 2013

E.g.f.: exp(x)*(4x^2 + 4x). - Geoffrey Critzer, Oct 07 2013

MATHEMATICA

Table[(2n)^2, {n, 0, 46}] (* Alonso del Arte, Apr 26 2011 *)

PROG

(MAGMA) [(2*n)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

(Maxima) makelist((2*n)^2, n, 0, 20); /* Martin Ettl, Jan 22 2013 */

(Haskell)

a016742 = (* 4) . (^ 2)

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

-- Reinhard Zumkeller, Jun 28 2015, Apr 20 2015

(PARI) a(n)=4*n^2 \\ Charles R Greathouse IV, Jul 28 2015

CROSSREFS

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

Cf. sequences listed in A254963.

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

Sequence in context: A281795 A063540 A055808 * A221285 A121317 A238259

Adjacent sequences:  A016739 A016740 A016741 * A016743 A016744 A016745

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified September 26 15:27 EDT 2017. Contains 292531 sequences.