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A016789 a(n) = 3*n + 2. 115

%I

%S 2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,

%T 74,77,80,83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,

%U 131,134,137,140,143,146,149,152,155,158,161,164,167,170,173,176,179

%N a(n) = 3*n + 2.

%C 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

%C 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.

%C Except for 2, exponents e such that x^e + x + 1 is reducible. - _N. J. A. Sloane_, Jul 19 2005

%C a(n) = A125199(n+1,1). - _Reinhard Zumkeller_, Nov 24 2006

%C 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

%C Union of A165334 and A165335. - _Reinhard Zumkeller_, Sep 17 2009

%C a(n) is the set of numbers congruent to{2,5,8} mod 9. - _Gary Detlefs_, Mar 07 2010

%C 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

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

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

%C A089911(2*a(n)) = 9. - _Reinhard Zumkeller_, Jul 05 2013

%C Also indices of even Bell numbers (A000110). - _Enrique Pérez Herrero_, Sep 10 2013

%C Central terms of the triangle A108872. - _Reinhard Zumkeller_, Oct 01 2014

%C A092942(a(n)) = 1 for n > 0. - _Reinhard Zumkeller_, Dec 13 2014

%C 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

%C Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - _Ron Knott_, Sep 16 2016

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

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

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

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 9.

%H L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=937">Encyclopedia of Combinatorial Structures 937</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

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

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

%F Sum_{n>=1} (-1)^n/a(n) = 1/3(Pi/sqrt(3) - log(2)). - _Benoit Cloitre_, Apr 05 2002

%F 1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - _Gary W. Adamson_, Dec 16 2006

%F Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... . [Jolley p. 48 eq (263)]

%F a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - _Philippe Deléham_, Nov 03 2008

%F a(n) = 6*n - a(n-1) + 1 with a(0)=2. - _Vincenzo Librandi_, Aug 25 2010

%p seq(3*n+2, n = 0 .. 50); # _Matt C. Anderson_, May 18 2017

%t Range[2, 500, 3] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)

%o (Haskell)

%o a016789 = (+ 2) . (* 3) -- _Reinhard Zumkeller_, Jul 05 2013

%o (PARI) vector(100,n,3*n-1) \\ _Derek Orr_, Apr 13 2015

%o (MAGMA) [3*n+2: n in [0..80]]; // _Vincenzo Librandi_, Apr 14 2015

%Y First differences of A005449.

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

%Y Cf. A087370.

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

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

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Last modified June 24 11:32 EDT 2017. Contains 288697 sequences.