Search: id:A001105 Results 1-1 of 1 results found. %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, More information %H A001105 Wolfram Klehr, Title? %H A001105 V. Ladma, Magic Numbers %H A001105 Index entries for sequences related to Chebyshev polynomials. %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 %O A001105 0,2 %A A001105 Bernd.Walter(AT)frankfurt.netsurf.de Search completed in 0.002 seconds