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A016825
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Positive integers congruent to 2 (mod 4): a(n) = 4*n+2, for n >= 0.
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211
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2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230, 234
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OFFSET
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0,1
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COMMENTS
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Twice the odd numbers, also called singly even numbers.
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.
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
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
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
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
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
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) = 4*n + 2, for n >= 0.
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) = 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
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EXAMPLE
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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).
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MAPLE
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a := n -> 4*n+2: seq(a(n), n = 0 .. 70); # Stefano Spezia, Jun 17 2019
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MATHEMATICA
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CoefficientList[Series[2*(1+x)/(1-x)^2, {x, 0, 70}], x] (* Eric W. Weisstein, Dec 01 2017 *)
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PROG
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(Magma) [4*n+2 : n in [0..70]];
(PARI) a(n)= 4*n+2
(Haskell)
a016825 = (+ 2) . (* 4)
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CROSSREFS
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Essentially the complement of A042965.
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KEYWORD
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nonn,cofr,easy,nice
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AUTHOR
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STATUS
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approved
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