%I A001105
%S A001105 0,2,8,18,32,50,72,98,128,162,200,242,288,338,392,450,512,578,648,722,
%T A001105 800,882,968,1058,1152,1250,1352,1458,1568,1682,1800,1922,2048,2178,
%U A001105 2312,2450,2592,2738,2888,3042,3200,3362,3528,3698,3872,4050,4232,4418
%N A001105 2*n^2.
%C A001105 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
%C A001105 "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.
%C A001105 These numbers also occur as the limiting periods in the Harmonic Periodic
Table of Gutierrez Samanez. See also the Klehr link.
%C A001105 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
%C A001105 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
%C A001105 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
%C A001105 Twice squares. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
%C A001105 a(n)=A016742(n)/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun
20 2008
%C A001105 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]
%C A001105 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]
%C A001105 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]
%D A001105 A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%D A001105 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.
%D A001105 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.
%D A001105 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer
Press, NY, 1950, p. 36.
%D A001105 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p.
44.
%D A001105 A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p.
213.
%H A001105 Julio Antonio Gutierrez Samanez, <a href="http://www.harras.be/hvar/kutiry/
index1.htm">More information</a>
%H A001105 Wolfram Klehr, <a href="http://www.apsidium.con/number/formula.xls">Title?</
a>
%H A001105 V. Ladma, <a href="http://www.sweb.cz/vladimir_ladma/english/notes/texts/
magicn.htm">Magic Numbers</a>
%H A001105 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001105 1/2 + 1/8 + 1/18 + 1/32 +...=(Pi)^2/12 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 21 2006
%F A001105 a(n)=A049452(n)-A033991(n), example:18=51-33, .. 210-138=72, etc... -
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007
%F A001105 a(n)=A000290(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
%F A001105 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]
%F A001105 a(n)=4*n+a(n-1)-6 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 08 2009]
%e A001105 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]
%e A001105 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]
%p A001105 a:=n->sum(n/2, j=1..n): seq(a(2*n), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 30 2007
%p A001105 with(finance):seq(add(futurevalue(n,1,2),k=1..n)/2,n=0..47); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%p A001105 with(finance):seq(add(cashflows([2,n,n], 0 ),k=0..n),n=-1..45); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008
%p A001105 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]
%t A001105 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]
%Y A001105 a(n) = ((-1)^(n+1))*A053120(2*n, 2).
%Y A001105 a(n) = A100345(n, n).
%Y A001105 Cf. A000290, A016742, A116471.
%Y A001105 Sequence in context: A055044 A067051 A074629 this_sequence A051787 A081324
A050804
%Y A001105 Adjacent sequences: A001102 A001103 A001104 this_sequence A001106 A001107
A001108
%K A001105 nonn,new
%O A001105 0,2
%A A001105 Bernd.Walter(AT)frankfurt.netsurf.de
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