0,2

4 times the squares.

Number of edges in the complete bipartite graph of order 5n, K_{n,4n} - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), 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 (njas(AT)research.att.com), 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 (info(AT)polprimos.com), 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 (see link). 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 Gutierrez Samanez (jgutierrezsamanez(AT)yahoo.com), 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. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 31 2009]

Number of cycles of length 3 in the king's graph associated with an (n+1) X (n+1) chessboard. [From 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 [From Ant King (mathstutoring(AT)ntlworld.com), Nov 04 2009]

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.

Index entries for sequences related to linear recurrences with constant coefficients

Julio Antonio Gutierrez Samanez Nuevo modelo matematico de la tabla periodica.

Various, Electron Configuration (Discussion in Physics Forums)

Wikipedia, Aufbau principle

O.g.f.: 4x(1+x)/(1-x)^3. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 28 2008

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

a(n)=8*n+a(n-1)-12 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]

a(46)=8464

For n=2, a(2)=8*2+0-12=4; n=3, a(3)=8*3+4-12=16; n=4, a(4)=8*4+16-12=36 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]

a:=n->sum(n, j=1..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

with(finance):seq(add(futurevalue(n, 1, 2), k=1..n), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

lst={}; Do[AppendTo[lst, n^2], {n, 0, 5!, 2}]; lst ...and/or... s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 3, 6!, 8}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]

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

Cf. A007742, A033991.

Sequence in context: A044065 A063540 A055808 this_sequence A121317 A063755 A166721

Adjacent sequences: A016739 A016740 A016741 this_sequence A016743 A016744 A016745

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

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