Search: id:A016789
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%I A016789
%S A016789 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 A016789 74,77,80,83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,
%U A016789 131,134,137,140,143,146,149,152,155,158,161,164,167,170,173,176,179
%N A016789 3n+2.
%C A016789 Except for 1, n such that sum(k=1,n,(k mod 3)*C(n,k)) is a power of 2 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 17 2002
%C A016789 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 (pbarry(AT)wit.ie), 
               Jan 28 2004. Artur Jasinski (grafix(AT)csl.pl), Dec 11 2007, remarks 
               that this should read Table[(3n + 2)(1 + Cos[2Pi*(3n + 2)/3 + Pi/
               3] - Sqrt[3] Sin[2Pi*(3n + 2)/3 + Pi/3])/3, {n, 0,20}] .
%C A016789 Except for 2, exponents e such that x^e+x+1 is reducible.
%C A016789 a(n) = A125199(n+1,1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 24 2006
%C A016789 Primitive roots of 3. - Paolo P. Lava (ppl(AT)spl.at), Feb 29 2008
%C A016789 The trajectory of these numbers under iteration of sum of cubes of digits 
               eventually turns out to be 371 or 407 (47 is the 1st of the second 
               kind) [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009]
%C A016789 If X=(28+n^3)/9, Y=(28+n^3)/3, Z=(n^4+28n)/9, then X^3+Y^3+Z^3=(X*Y)^2 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 30 2009]
%C A016789 Union of A165334 and A165335. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 17 2009]
%D A016789 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 
               16.
%D A016789 Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269
%H A016789 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016789 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
               Observatio de summis divisorum</a> p. 9.
%H A016789 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation 
               on the sums of divisors</a> p. 9.
%H A016789 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A016789 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=937">
               Encyclopedia of Combinatorial Structures 937</a>
%H A016789 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")
%F A016789 G.f.: (2+x)/(1-x)^2. a(n)=3+a(n-1).
%F A016789 sum(n=1, inf, (-1)^n/a(n))=1/3(Pi/sqrt(3)-ln(2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002
%F A016789 1/2 - 1/5 + 1/8 - 1/11...= (1/3)*(Pi/sqrt(3) - ln 2). [Jolley] - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2006
%F A016789 a(n)=2*a(n-1)-a(n-2); a(0)=2, a(1)=5. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2008]
%F A016789 a(n)=6*n-a(n-1)-5 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 23 2009]
%e A016789 For n=2, X=4, Y=12, Z=8, 4^3+12^3+8^3=(4*12)^2: n=5, X=17, Y=51, Z=85, 
               17^3+51^3+85^3=(17*51)^2: n=8, X=60, Y=180, Z=480, 60^3+180^3+480^3=(60*180)^2 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jun 30 2009]
%e A016789 For n=2, a(2)=6*2-2-5=5; n=3, a(3)=6*3-5-5=8; n=4, a(4)=6*4-8-5=11 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 23 2009]
%p A016789 [seq(4*binomial(3*n, 2)/binomial(2*n,1)/3, n=1..60)]; - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jan 16 2007
%p A016789 a[1]:=-1:for n from 2 to 100 do a[n]:=a[n-1]+3 od: seq(a[n], n=2..61); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%t A016789 f[n_]:=3*n+2; lst={};Do[a=f[n];AppendTo[lst,a],{n,0,6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
%o A016789 (Other) sage: [i+2 for i in range(180) if gcd(i,3) == 3] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A016789 (Other) sage: [crt(2, n, 3, 2) for n in xrange(2, 62)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2009]
%Y A016789 A016789(n)=1+A016777(n).
%Y A016789 First differences of A005449.
%Y A016789 a(n)=A124388(n)/9.
%Y A016789 Cf. A002939, A017041, A017485, A125202.
%Y A016789 Cf. A017233.
%Y A016789 Sequence in context: A109232 A064718 A078608 this_sequence A165334 A135677 
               A163516
%Y A016789 Adjacent sequences: A016786 A016787 A016788 this_sequence A016790 A016791 
               A016792
%K A016789 nonn,easy,new
%O A016789 0,1
%A A016789 N. J. A. Sloane (njas(AT)research.att.com).

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