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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

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

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

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

Right-hand side of the binomial coefficient identity Sum_{k = 0..2*n} (-1)^(k+1)* binomial(2*n,k)*binomial(2*n + k,k)*(2*n - k) = a(n). - Peter Bala, Jan 12 2022

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.
• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..900
• R. P. Boas and N. J. A. Sloane, Correspondence, 1974.
• Leo Tavares, Illustration: X Squares
• 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).

Crossrefs

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

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

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.

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.

Cf. sequences listed in A254963.

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

Cf. A000384, A014105.

Cf. A016813.

Programs

GAP

List([0..100], n -> (2*n)^2); # Muniru A Asiru, Jan 28 2018

Haskell

a016742 = (* 4) . (^ 2)
a016742_list = 0 : map (subtract 4) (zipWith (+) a016742_list [8, 16 ..])
-- Reinhard Zumkeller, Jun 28 2015, Apr 20 2015

Magma

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

Maple

seq((2*n)^2, n=0..100); # Muniru A Asiru, Jan 28 2018

Mathematica

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

Maxima

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

PARI

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

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

Sum_{n>=1} 1/a(n) = (1/4)*Pi^2/6 = Pi^2/24. - Ant King, Nov 04 2009

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*Product_{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

a(n) = A000384(n) + A014105(n). - Bruce J. Nicholson, Nov 11 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 (A245058). - Amiram Eldar, Oct 10 2020

From Amiram Eldar, Jan 25 2021: (Start)

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

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

a(n) = A016754(n) - A016813(n). - Leo Tavares, Feb 24 2022

Extensions

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