

A016789


a(n) = 3*n + 2.


184



2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2.  Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3)  sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1x^3)^2.  Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3)  sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible.  N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind).  Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9.  Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k.  Gary Detlefs, Mar 08 2010
a(n1), n >= 1, is also the complex dimension of the manifold E(S), the set of all secondorder irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149.  Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n1) + x^n is reducible.  Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640.  Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z).  Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777.  Michael Somos, May 27 2019
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information.  Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts.  Douglas Boffey, Aug 01 2022 [See the Wikipedia link.  Wolfdieter Lang, Sep 29 2022]
a(n2) is the maximum sum of the span (or L(2,1)labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(31) = 5.  Allan Bickle, Apr 20 2023


REFERENCES

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269


LINKS

D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146159; see also the arXiv version, arXiv:math/0411140 [math.NT], 20042005.


FORMULA

G.f.: (2+x)/(1x)^2.
a(n) = 3 + a(n1).
Sum_{n>=1} (1)^n/a(n) = (1/3)*(Pi/sqrt(3)  log(2)).  Benoit Cloitre, Apr 05 2002
1/2  1/5 + 1/8  1/11 + ... = (1/3)*(Pi/sqrt(3)  log 2). [Jolley]  Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3)  log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1).  Gilian Breysens, Jul 21 2017


EXAMPLE

G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ...  Michael Somos, May 27 2019


MAPLE



MATHEMATICA

LinearRecurrence[{2, 1}, {2, 5}, 70] (* Harvey P. Dale, Aug 11 2021 *)


PROG

(Haskell)
(PARI) vector(100, n, 3*n1) \\ Derek Orr, Apr 13 2015
(Python) for n in range(0, 100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018


CROSSREFS

Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. similar sequences with closed form (2*k1)*n+k listed in A269044.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



