A016789 a(n) = 3*n + 2.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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)/(1-x^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

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 = k*n + 2 for integer k. - Gary Detlefs, Mar 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

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

Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021

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(n-2) 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(3-1) = 5. - Allan Bickle, Apr 20 2023

Examples

			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
		

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

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
• H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
• Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
• 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
• L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
• 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")
• Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Leo Tavares, Illustration: Capped Triangular Frames
• Wikipedia, Sprouts (game)
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

First differences of A005449.

Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.

Cf. A087370.

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

Cf. A000096, A000217.

Cf. A016933, A016969.

Programs

GAP

List([0..70],n->3*n+2); # Muniru A Asiru, Nov 02 2018

Haskell

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

Magma

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

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

PARI

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

Python

for n in range(0,100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018

Formula

G.f.: (2+x)/(1-x)^2.

a(n) = 3 + a(n-1).

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

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

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

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)]

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

Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017

E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018

a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019

a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019

a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021

a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021