Search: id:A005843
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%I A005843 M0985
%S A005843 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,
%T A005843 46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,
%U A005843 88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120
%N A005843 The even numbers: a(n) = 2n.
%C A005843 -2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann 
               zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
%C A005843 Numbers of carbons in the oligomers of acetylene. - Parthasarathy Nambi 
               (PachaNambi(AT)yahoo.com), Oct 27 2005
%C A005843 Excluding the first two terms, this sequenc is the same as the number 
               of carbons in the oligomers of cyclobutane sharing a common edge. 
               - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 05 2006
%C A005843 If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then 
               a(n-2) is the number of 2-subsets of X intersecting both Y and Z. 
               - Milan R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A005843 A134452(a(n)) = 0; A134451(a(n)) = 2 for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 27 2007
%C A005843 Omitting the initial zero gives the number of prime divisors with multiplicity 
               of product of terms of n-th row of A077553. - Ray Chandler (rayjchandler(AT)sbcglobal.net), 
               Aug 21 2003
%C A005843 A059841(a(n))=1, A000035(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 29 2008]
%C A005843 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Oct 28 2008: 
               (Start)
%C A005843 (APSO) Alternating partial sums of
%C A005843 (a-b+c-d+e-f+g...)=(a+b+c+d+e+f+g...)-2*(b+d+f...)
%C A005843 it appears that APSO A005843 =
%C A005843 A052928 = A002378 - 2*(A116471)
%C A005843 A116471=2*A008794
%C A005843 (End)
%C A005843 If A=[A157872] 9*n.^2-3 (6, 33, 78, 141,.,); Y=[A005843] 2*n (except 
               the first term , 2,4,6,8,.,); X=[A140811] 6*n^2-1 (except the first 
               term, 5,23,5395,.,), we have, for all terms, Pell's equation X^2-A*Y^2=1. 
               Example: 5^2-6 *2^2=1; 23^2-33*4^2=1; 53^2-78*6^2=1. [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]
%C A005843 A056753(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 23 2009]
%D A005843 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 2.
%D A005843 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005843 N. J. A. Sloane, <a href="b005843.txt">Table of n, a(n) for n = 0..10000</
               a>
%H A005843 Kevin Fagan, <a href="http://chesswanks.com/pot/IntelligenceTest.jpg">
               Drabble cartoon, Jun 15 1987: Intelligence Test</a>
%H A005843 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005843 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A005843 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">
               X^2-AY^2=1</a> [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%H A005843 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005843 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005843 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EvenNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A005843 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HamiltonianCircuit.html">Hamiltonian Circuit</a>
%H A005843 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>.
%H A005843 Edward Everett Withford, <a href="http://quod.lib.umich.edu/cgi/t/text/
               text-idx?c=umhistmath;cc=umhistmath;idno=abv2773.0001.001;view=toc">
               Pell Equation</a> [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%H A005843 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A005843 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A005843 G.f.: 2*x/(1-x)^2.
%F A005843 Inverse binomial transform of A036289, n*2^n. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 
               Jan 13 2006
%F A005843 a(0)=0, a(1)=2, a(n)=2a(n-1)-a(n-2). - Jaume Oliver i Lafont (joliverlafont(AT)gmail.com), 
               May 07 2008
%F A005843 a(n)=Sum{k=1,n}floor(6n/4^k+1/2) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Jun 04 2009]
%F A005843 a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) 
               = A005408(n) - 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Sep 05 2009]
%p A005843 A005843 := n->2*n;
%p A005843 A005843:=2/(z-1)**2; [S. Plouffe in his 1992 dissertation.]
%t A005843 lst={};Do[AppendTo[lst, 2*n], {n, 6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Aug 29 2008]
%o A005843 (MAGMA) [ 2*n : n in [0..100]];
%o A005843 (R) seq(0,200,2)
%o A005843 (PARI) A005843(n) = 2*n
%Y A005843 Cf. A000027, A005408.
%Y A005843 Cf. A001358, A005843, A077553, A077554, A077555, A002024, A087112.
%Y A005843 Cf. A157872, A140811 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Mar 08 2009]
%Y A005843 Sequence in context: A087113 A004275 A119432 this_sequence A076032 A004277 
               A122080
%Y A005843 Adjacent sequences: A005840 A005841 A005842 this_sequence A005844 A005845 
               A005846
%K A005843 easy,core,nonn,nice
%O A005843 0,2
%A A005843 N. J. A. Sloane (njas(AT)research.att.com).

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