Search: id:A016825
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%I A016825
%S A016825 2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,94,
%T A016825 98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,162,166,
%U A016825 170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230,234
%N A016825 Numbers congruent to 2 mod 4: a(n) = 4n+2.
%C A016825 Continued fraction for (e-1)/(e+1).
%C A016825 No solutions to a(n)=b^2-c^2 - Henry Bottomley (se16(AT)btinternet.com), 
               Jan 13 2001
%C A016825 Apart from initial term(s), dimension of the space of weight 2n cuspidal 
               newforms for Gamma_0( 70 ).
%C A016825 Sequence gives n such that 8 is the largest power of 2 dividing A003629(k)^n-1 
               for any k - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002
%C A016825 n such that sum(d|n,(-1)^d)=A048272(n)=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 15 2002
%C A016825 Also n such that sum(d|n,phi(d)*mu(n/d))=A007431(n)=0 - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Apr 15 2002
%C A016825 Also n such that sum(d|n,d/AOOOO5(d)*mu(n/d))=0, n such that sum(d|n,
               AOOOO5(d)/d*mu(n/d))=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 19 2002
%C A016825 Solutions to phi[x]=phi[x/2]; primorial numbers are here. - Labos E. 
               (labos(AT)ana.sote.hu), Dec 16 2002
%C A016825 Together with 1, numbers that are not the leg of a primitive Pythagorean 
               triangle. - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 25 2003
%C A016825 Numbers having equal numbers of odd and even divisors: A001227(a(n))=A000005(2*a(n)). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 28 2003
%C A016825 Maximum number of electrons in an atomic subshell with orbital quantum 
               number l is 4l+2.
%C A016825 For n>0: complement of A107750 and A023416(a(n)-1)=A023416(a(n))<>A023416(a(n)+1). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 23 2005
%C A016825 Also the minimal value of sum([p(i)-p(i+1)]^2, i=1..n+2), where p(n+3)=p(1), 
               as p ranges over all permutations of {1,2,...,n+2} (see the Mihai 
               reference). Example: a(2)=10 because the values of the sum for the 
               permutations of {1,2,3,4}are 10 (8 times), 12 (8 times) and 18 (8 
               times). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 30 2005
%C A016825 Except for a(n)=2, numbers having 4 as an anti-divisor. - Alexandre Wajnberg 
               (alexandre.wajnberg(AT)skynet.be), Oct 02 2005
%C A016825 This is also the number of polyacenes in carbon nanotubes. See page 413 
               equation 12 of the paper by I. Lukovits and D. Janezic. - Parthasarathy 
               Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
%C A016825 A139391(a(n)) = A006370(a(n)) = A005408(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 17 2008
%C A016825 Twice odd numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 16 
               2008]
%C A016825 Terms of a(n) in A102261= 2, 6, 6, 10, 14, 10, 10, 14, 14, 22, 26 . [From 
               Paul Curtz (bpcrtz(AT)free.fr), Sep 07 2008]
%C A016825 Also a(n) = (n-1) + n + (n+1) + (n+2), so a(n) and -a(n) are all the 
               integers that are sums of four consecutive integers. [From Rick L. 
               Shepherd (rshepherd2(AT)hotmail.com), Mar 21 2009]
%C A016825 (e-1)/(e+1) = tanh(1/2) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 09 2009]
%D A016825 A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%D A016825 J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
%D A016825 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", 
               J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004).
%D A016825 V. Mihai, Problem 10725, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
%D A016825 H. Bass, Mathematics, Mathematicians and Mathematics Education, Bull. 
               Amer. Math. Soc. (N.S.) 42 (2004), no. 4, 417-430.
%H A016825 Harry J. Smith, <a href="b016825.txt">Table of n, a(n) for n=0,...,20000</
               a>
%H A016825 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016825 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A016825 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0new.gp">
               Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A016825 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The 
               modular forms database</a>
%H A016825 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SinglyEvenNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A016825 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareNumber.html">Square Number</a>
%H A016825 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
               Contfrac</a>
%H A016825 <a href="Sindx_Con.html#confC">Index entries for continued fractions 
               for constants</a>
%F A016825 a(n)=2*A005408(n) - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 28 2003
%F A016825 a(n) = A118413(n+1,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 27 2006
%F A016825 G.f.: 2* (1+x)/(1-x)^2. E.g.f.: 2*(1+2*x)*exp(x). a(n)= a(n-1) + 4. a(-1-n)= 
               -a(n). - Michael Somos Apr 11 2007
%F A016825 a(n)=8*n-a(n-1)-8 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 24 2009]
%e A016825 0.4621171572600097585023184... = 0 + 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))) 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]
%e A016825 For n=2, a(2)=8*2-2-8=6; n=3, a(3)=8*3-6-8=10; n=4, a(4)=8*4-10-8=14 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 24 2009]
%p A016825 with(finance):seq(add(cashflows([0,0,4], 0 ),k=1..n)+2,n=0..58); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008
%t A016825 lst={};Do[AppendTo[lst, 4*n+2], {n, 6!}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Aug 29 2008]
%o A016825 (MAGMA) [4*n+2 : n in [0..100] ];
%o A016825 (PARI) {a(n)= 4*n+2}
%o A016825 (PARI) { allocatemem(932245000); default(realprecision, 180000); x=contfrac(tanh(1/
               2)); for (n=2, 20002, write("b016825.txt", n-2, " ", x[n])); } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 09 2009]
%o A016825 (Other) sage: [i+2 for i in range(236) if gcd(i,4) == 4] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A016825 (Other) sage: [crt(2, n, 4,3 ) for n in xrange(2, 61)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 07 2009]
%Y A016825 Cf. A107687. First differences of A001105.
%Y A016825 Cf. A160327 = Decimal expansion. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 09 2009]
%Y A016825 Sequence in context: A068977 A111284 A130824 this_sequence A161718 A122905 
               A132417
%Y A016825 Adjacent sequences: A016822 A016823 A016824 this_sequence A016826 A016827 
               A016828
%K A016825 nonn,easy,nice,cofr,new
%O A016825 0,1
%A A016825 N. J. A. Sloane (njas(AT)research.att.com).

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