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A016789 a(n) = 3*n + 2. 115
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)*C(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(2Pi*n/3 + Pi/3) - sqrt(3)sin(2Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004. Artur Jasinski, Dec 11 2007, remarks that this should read (3n + 2)(1 + Cos[2Pi*(3n + 2)/3 + Pi/3] - Sqrt[3] Sin[2Pi*(3n + 2)/3 + Pi/3])/3, or in Maple format (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

a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006

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

Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 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 = kn + 2 for integer k. - Gary Detlefs, Mar 08 2010

Except for the first term, a(n) = ceil(A179896 / n) for n > 0 and remainder != 0. - Odimar Fabeny, Sep 08 2010

These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012

A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013

Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013

Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014

A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014

a(n-1), n >=1, is also the complex dimension of the manifold E(S), the set of all second order 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^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016

REFERENCES

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 16.

Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

LINKS

Table of n, a(n) for n=0..59.

L. Euler, Observatio de summis divisorum p. 9.

L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937

Tanya Khovanova, Recursive Sequences

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

G.f.: (2+x)/(1-x)^2. a(n) = 3 + a(n-1).

a(n) = 1 + A016777(n).

a(n) = A124388(n)/9.

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... . [Jolley p. 48 eq (263)]

a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008

a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010

MAPLE

seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017

MATHEMATICA

Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

PROG

(Haskell)

a016789 = (+ 2) . (* 3)  -- Reinhard Zumkeller, Jul 05 2013

(PARI) vector(100, n, 3*n-1) \\ Derek Orr, Apr 13 2015

(MAGMA) [3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015

CROSSREFS

First differences of A005449.

Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388.

Cf. A087370.

Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

Sequence in context: A078608 A189934 A189386 * A190082 A165334 A189512

Adjacent sequences:  A016786 A016787 A016788 * A016790 A016791 A016792

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 24 11:11 EDT 2017. Contains 288697 sequences.