OFFSET
0,1
COMMENTS
Twice the odd numbers, also called singly even numbers.
Numbers having equal numbers of odd and even divisors: A001227(a(n)) = A000005(2*a(n)). - Reinhard Zumkeller, Dec 28 2003
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.
No solutions to a(n) = b^2 - c^2. - Henry Bottomley, Jan 13 2001
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
k such that Sum_{d|k} (-1)^d = A048272(k) = 0. - Benoit Cloitre, Apr 15 2002
Also k such that Sum_{d|k} phi(d)*mu(k/d) = A007431(k) = 0. - Benoit Cloitre, Apr 15 2002
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
Solutions to phi(x) = phi(x/2); primorial numbers are here. - Labos Elemer, Dec 16 2002
Together with 1, numbers that are not the leg of a primitive Pythagorean triangle. - Lekraj Beedassy, Nov 25 2003
For n > 0: complement of A107750 and A023416(a(n)-1) = A023416(a(n)) <> A023416(a(n)+1). - Reinhard Zumkeller, May 23 2005
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
Except for a(n)=2, numbers having 4 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
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
The denominator in Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 13 2011
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
Numbers k such that Sum_{j=1..k} j^3 is not a multiple of k. - Chai Wah Wu, Aug 23 2017
Numbers k such that Lucas(k) is a multiple of 3. - Bruno Berselli, Oct 17 2017
Also numbers k such that t^k == -1 (mod 5), where t is a term of A047221. - Bruno Berselli, Dec 28 2017
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
Also numbers ending with 10 in base 2. - John Keith, May 09 2022
REFERENCES
H. Bass, Mathematics, Mathematicians and Mathematics Education, Bull. Amer. Math. Soc. (N.S.) 42 (2004), no. 4, 417-430.
Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..20000
Tanya Khovanova, Recursive Sequences
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
I. Lukovits and D. Janezic, Enumeration of conjugated circuits in nanotubes, J. Chem. Inf. Comput. Sci. 44 (2004), 410-414.
Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
Paolo Emilio Ricci, Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree, Symmetry (2018) Vol. 10, No. 12, 671.
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Singly Even Number
Eric Weisstein's World of Mathematics, Square Number
G. Xiao, Contfrac
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n) = 4*n + 2, for n >= 0.
a(n) = 2*A005408(n). - Lekraj Beedassy, Nov 28 2003
a(n) = A118413(n+1,2) for n>1. - Reinhard Zumkeller, Apr 27 2006
From Michael Somos, Apr 11 2007: (Start)
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). (End)
a(n) = 8*n - a(n-1) for n > 0, a(0)=2. - Vincenzo Librandi, Nov 20 2010
From Reinhard Zumkeller, Jun 11 2012, Jun 30 2012 and Jul 20 2012: (Start)
A080736(a(n)) = 0.
A007814(a(n)) = 1;
A037227(a(n)) = 3.
A214546(a(n)) = 0. (End)
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
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2-sqrt(2)) (A285871).
Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(1-1/sqrt(2)) (A154739). (End)
EXAMPLE
0.4621171572600097585023184... = 0 + 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))), i.e., c.f. for tanh(1/2).
2.1639534137386528487700040... = 2 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + ...)))), i.e., c.f. for coth(1/2).
MAPLE
a := n -> 4*n+2: seq(a(n), n = 0 .. 70); # Stefano Spezia, Jun 17 2019
MATHEMATICA
Range[2, 280, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
4*Range[0, 70] +2 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{2, -1}, {2, 6}, 70] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[2*(1+x)/(1-x)^2, {x, 0, 70}], x] (* Eric W. Weisstein, Dec 01 2017 *)
NestList[#+4&, 2, 60] (* Harvey P. Dale, Apr 08 2022 *)
PROG
(Magma) [4*n+2 : n in [0..70]];
(PARI) a(n)= 4*n+2
(PARI) contfrac(tanh(1/2)) \\ To illustrate the 3rd comment. - Harry J. Smith, May 09 2009 [Edited by M. F. Hasler, Mar 09 2020]
(Haskell)
a016825 = (+ 2) . (* 4)
a016825_list = [2, 6 ..] -- Reinhard Zumkeller, Feb 14 2012
(GAP) Flat(List([0..70], n->4*n+2)); # Stefano Spezia, Jun 17 2019
(Sage) [4*n+2 for n in (0..70)] # G. C. Greubel, Jun 28 2019
CROSSREFS
KEYWORD
nonn,cofr,easy,nice,changed
AUTHOR
STATUS
approved