0,2

Number of edges of the complete bipartite graph of order 3n, K_{n,2n}. - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002

"If each period in the periodic system ends in a rare gas ..., the number of elements in a period can be found from the ordinal number n of the period by the formula: L = ((2n+3+(-1)^n)^2)/8..." - Nature Jun 09 1951; Nature 411 (Jun 07 2001), p. 648. This produces the present sequence doubled up.

These numbers also occur as the limiting periods in the Harmonic Periodic Table of Gutierrez Samanez. See also the Klehr link.

Let z(1)=I (I^2=-1), z(k+1) = 1/(z(k)+2I); then a(n)=(-1)*Imag(z(n+1))/real(z(n+1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2002

Maximum number of electrons in an atomic shell with total quantum number n. Partial sums of A016825. - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Dec 19 2004

Arithmetic mean of triangular numbers in pairs: (1+3)/2, (6+10)/2,(15+21)/2,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2005

Twice squares. - Omar E. Pol (info(AT)polprimos.com), May 14 2008

a(n)=A016742(n)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

Except for the first term, numbers n such that (9*n^3)/8 is a square. Example: (9*2^3)/8=9=3^2; (9*8^3)/8=576=24^2; (9*18^3)/8=6561=81^2; (9*32^3)/8=36864=192^2; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 16 2009]

Also, except for the first term, numbers n such that 72*n^3 is a square. Example: 72*2^3=24^2; 72*8^3=192^2; 72*18^3=648^2 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

Integral areas of isosceles right triangles with rational legs (Legs are 2n and triangles are nondegenerate for n>0). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 29 2009]

A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.

Martin Gardner, The Colossal Book of Mathematics, Classic Puzzles, Paradoxes and Problems, Chapter 2 entitled "The Calculus of Finite Differences," W. W. Norton and Company, New York, 2001, pages 12-13.

Julio Antonio Gutierrez Samanez, "Sistema Periodico Armonico y leyes Geneticas de los Elementos Quimicos" (Harmonic Periodic System and Genetic Laws of Chemical Elements), Cusco, Peru 2004. ISBN: 9972-33-063-X.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 44.

A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.

Julio Antonio Gutierrez Samanez, More information

Wolfram Klehr, Title?

V. Ladma, Magic Numbers

Index entries for sequences related to Chebyshev polynomials.

1/2 + 1/8 + 1/18 + 1/32 +...=(Pi)^2/12 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006

a(n)=A049452(n)-A033991(n), example:18=51-33, .. 210-138=72, etc... - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007

a(n)=A000290(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008

Let this A001105=F(actor) then F*8=Q^2 always. Q=4*n if n>=0 and n are the unique number of exact roots Q. [From david scheers (dscheers(AT)webpoint.nl), Mar 15 2009]

a(n)=4*n+a(n-1)-6 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

example: F=50. n=5. Q=20. 20^2=50*8=400 Q F= this or next Q F is root? n 2*n^2 Q= n*4 0 *2^3= 0 0 0 0 0 1 *2^3= 8 no 0 0 2 *2^3= 16 yes! 1 2 4 8 *2^3= 64 yes! 2 8 8 18 *2^3= 144 yes! 3 18 12 32 *2^3= 256 yes! 4 32 16 50 *2^3= 400 yes! 5 50 20 72 *2^3= 576 yes! 6 72 24 98 *2^3= 784 yes! 7 98 28 128 *2^3= 1024 yes! 8 128 32 162 *2^3= 1296 yes! 9 162 36 200 *2^3= 1600 yes! 10 200 40 242 *2^3= 1936 yes! 11 242 44 288 *2^3= 2304 yes! 12 288 48 338 *2^3= 2704 yes! 13 338 52 392 *2^3= 3136 yes! 14 392 56 [From david scheers (dscheers(AT)webpoint.nl), Mar 15 2009]

For n=2, a(2)=4*2+0-6=2; n=3, a(3)=4*3+2-6=8; n=4, a(4)=4*4+8-6=18 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

a:=n->sum(n/2, j=1..n): seq(a(2*n), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

with(finance):seq(add(futurevalue(n, 1, 2), k=1..n)/2, n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

with(finance):seq(add(cashflows([2, n, n], 0 ), k=0..n), n=-1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008

a:=n->sum(2+sum(2, k=2..n), k=1..n):seq(a(n), n=0...43); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]

s=0; lst={s}; Do[s+=n+++2; AppendTo[lst, s], {n, 0, 7!, 4}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]

a(n) = ((-1)^(n+1))*A053120(2*n, 2).

a(n) = A100345(n, n).

Cf. A000290, A016742, A116471.

Sequence in context: A055044 A067051 A074629 this_sequence A051787 A081324 A050804

Adjacent sequences: A001102 A001103 A001104 this_sequence A001106 A001107 A001108

nonn

Bernd.Walter(AT)frankfurt.netsurf.de