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%I #231 Oct 10 2022 07:55:16
%S 2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,
%T 94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,
%U 162,166,170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230,234
%N Positive integers congruent to 2 (mod 4): a(n) = 4*n+2, for n >= 0.
%C Twice the odd numbers, also called singly even numbers.
%C Numbers having equal numbers of odd and even divisors: A001227(a(n)) = A000005(2*a(n)). - _Reinhard Zumkeller_, Dec 28 2003
%C Continued fraction for coth(1/2) = (e+1)/(e-1). The continued fraction for tanh(1/2) = (e-1)/(e+1) would be a(0) = 0, a(n) = A016825(n-1), n >= 1.
%C No solutions to a(n) = b^2 - c^2. - _Henry Bottomley_, Jan 13 2001
%C Sequence gives m such that 8 is the largest power of 2 dividing A003629(k)^m-1 for any k. - _Benoit Cloitre_, Apr 05 2002
%C k such that Sum_{d|k} (-1)^d) = A048272(k) = 0. - _Benoit Cloitre_, Apr 15 2002
%C Also k such that Sum_{d|k} phi(d)*mu(k/d) = A007431(k) = 0. - _Benoit Cloitre_, Apr 15 2002
%C Also k such that Sum_{d|k} (d/A000005(d))*mu(k/d) = 0, k such that Sum_{d|k}(A000005(d)/d)*mu(k/d) = 0. - _Benoit Cloitre_, Apr 19 2002
%C Solutions to phi(x) = phi(x/2); primorial numbers are here. - _Labos Elemer_, Dec 16 2002
%C Together with 1, numbers that are not the leg of a primitive Pythagorean triangle. - _Lekraj Beedassy_, Nov 25 2003
%C For n > 0: complement of A107750 and A023416(a(n)-1) = A023416(a(n)) <> A023416(a(n)+1). - _Reinhard Zumkeller_, May 23 2005
%C Also the minimal value of Sum_{i=1..n+2} (p(i) - p(i+1))^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_, Jul 30 2005
%C Except for a(n)=2, numbers having 4 as an anti-divisor. - _Alexandre Wajnberg_, Oct 02 2005
%C A139391(a(n)) = A006370(a(n)) = A005408(n). - _Reinhard Zumkeller_, Apr 17 2008
%C 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. - _Rick L. Shepherd_, Mar 21 2009
%C The denominator in Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - _Mohammad K. Azarian_, Oct 13 2011
%C This sequence gives the positive zeros of i^x + 1 = 0, x real, where i^x = exp(i*x*Pi/2). - _Ilya Gutkovskiy_, Aug 08 2015
%C Numbers k such that Sum_{j=1..k} j^3 is not a multiple of k. - _Chai Wah Wu_, Aug 23 2017
%C Numbers k such that Lucas(k) is a multiple of 3. - _Bruno Berselli_, Oct 17 2017
%C Also numbers k such that t^k == -1 (mod 5), where t is a term of A047221. - _Bruno Berselli_, Dec 28 2017
%C The even numbers form a ring, and these are the primes in that ring. Note that unique factorization into primes does not hold, since 60 = 2*30 = 6*10. - _N. J. A. Sloane_, Nov 11 2019
%C Also numbers ending with 10 in base 2. - _John Keith_, May 09 2022
%D H. Bass, Mathematics, Mathematicians and Mathematics Education, Bull. Amer. Math. Soc. (N.S.) 42 (2004), no. 4, 417-430.
%D Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
%D J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
%D Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).
%H Harry J. Smith, <a href="/A016825/b016825.txt">Table of n, a(n) for n = 0..20000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H D. H. Lehmer, <a href="/A016825/a016825.pdf">Continued fractions containing arithmetic progressions</a>, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
%H I. Lukovits and D. Janezic, <a href="http://dx.doi.org/10.1021/ci034240k">Enumeration of conjugated circuits in nanotubes</a>, J. Chem. Inf. Comput. Sci. 44 (2004), 410-414.
%H Vasile Mihai and Michael Woltermann, <a href="http://www.jstor.org/stable/2695399">Problem 10725: The Smoothest and Roughest Permutations</a>, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
%H Paolo Emilio Ricci, <a href="https://doi.org/10.3390/sym10120671">Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree</a>, Symmetry (2018) Vol. 10, No. 12, 671.
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SinglyEvenNumber.html">Singly Even Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 4*n + 2, for n >= 0.
%F a(n) = 2*A005408(n). - _Lekraj Beedassy_, Nov 28 2003
%F a(n) = A118413(n+1,2) for n>1. - _Reinhard Zumkeller_, Apr 27 2006
%F From _Michael Somos_, Apr 11 2007: (Start)
%F G.f.: 2*(1+x)/(1-x)^2.
%F E.g.f.: 2*(1+2*x)*exp(x).
%F a(n) = a(n-1) + 4.
%F a(-1-n) = -a(n). (End)
%F a(n) = 8*n - a(n-1) for n > 0, a(0)=2. - _Vincenzo Librandi_, Nov 20 2010
%F From _Reinhard Zumkeller_, Jun 11 2012, Jun 30 2012 and Jul 20 2012: (Start)
%F A080736(a(n)) = 0.
%F A007814(a(n)) = 1;
%F A037227(a(n)) = 3.
%F A214546(a(n)) = 0. (End)
%F a(n) = T(n+2) - T(n-2) where T(n) = n*(n+1)/2 = A000217(n). In general, if M(k,n) = 2*k*n + k, then M(k,n) = T(n+k) - T(n-k). - _Charlie Marion_, Feb 24 2020
%e 0.4621171572600097585023184... = 0 + 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))), i.e., c.f. for tanh(1/2).
%e 2.1639534137386528487700040... = 2 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + ...)))), i.e., c.f. for coth(1/2).
%p a := n -> 4*n+2: seq(a(n), n = 0 .. 70); # _Stefano Spezia_, Jun 17 2019
%t Range[2, 280, 4] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t 4*Range[0, 70] +2 (* _Eric W. Weisstein_, Dec 01 2017 *)
%t LinearRecurrence[{2, -1}, {2, 6}, 70] (* _Eric W. Weisstein_, Dec 01 2017 *)
%t CoefficientList[Series[2*(1+x)/(1-x)^2, {x,0,70}], x] (* _Eric W. Weisstein_, Dec 01 2017 *)
%t NestList[#+4&,2,60] (* _Harvey P. Dale_, Apr 08 2022 *)
%o (Magma) [4*n+2 : n in [0..70]];
%o (PARI) a(n)= 4*n+2
%o (PARI) contfrac(tanh(1/2)) \\ To illustrate the 3rd comment. - _Harry J. Smith_, May 09 2009 [Edited by _M. F. Hasler_, Mar 09 2020]
%o (Haskell)
%o a016825 = (+ 2) . (* 4)
%o a016825_list = [2, 6 ..] -- _Reinhard Zumkeller_, Feb 14 2012
%o (GAP) Flat(List([0..70], n->4*n+2)) # _Stefano Spezia_, Jun 17 2019
%o (Sage) [4*n+2 for n in (0..70)] # _G. C. Greubel_, Jun 28 2019
%Y Cf. A107687. First differences of A001105.
%Y Cf. A160327 (decimal expansion).
%Y Subsequence of A042963.
%Y Essentially the complement of A042965.
%K nonn,cofr,easy,nice
%O 0,1
%A _N. J. A. Sloane_
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