%I A016945
%S A016945 3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99,105,111,
%T A016945 117,123,129,135,141,147,153,159,165,171,177,183,189,195,
%U A016945 201,207,213,219,225,231,237,243,249,255,261,267,273,279
%N A016945 6n+3.
%C A016945 Apart from initial term(s), dimension of the space of weight 2n cuspidal 
               newforms for Gamma_0( 37 ).
%C A016945 Continued fraction expansion of tanh(1/3).
%C A016945 If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) 
               is the number of 3-subsets of X intersecting both Y and Z. - Milan 
               R. Janjic (agnus(AT)blic.net), Sep 08 2007
%C A016945 If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) 
               is the number of 3-subsets of X intersecting both Y and Z. - Milan 
               R. Janjic (agnus(AT)blic.net), Sep 19 2007
%C A016945 A008615(a(n)) = n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 27 2008
%C A016945 Leaves of the Odd Collatz-Tree: a(n) has no odd predecessors in all '3x+1' 
               trajectories where it occurs: A139391(2*k+1) <> a(n) for all k; A082286(n)=A006370(a(n)). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 17 2008
%C A016945 A157176(a(n)) = A103333(n+1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 24 2009]
%C A016945 Contribution from L Pearson (loren.pearson(AT)gmail.com), Jul 02 2009: 
               (Start)
%C A016945 Values of n in 2^n-1 that produce a composite with 7 as a factor.
%C A016945 Their distribution in 2^n-1 sequence equidistant between terms that have 
               multiple factors of 3 (n=6,12,18,24,30,36,... where the number of 
               factors of 3 equals to [number of times 3 divides n] + 1), recognizing 
               that all even n in the 2^n-1 sequence have at least one factor of 
               3.
%C A016945 Other odd n appear to be unrelated prime or semi-prime composites.
%C A016945 (End)
%D A016945 Friedrich L. Bauer, 'Der (ungerade) Collatz-Baum', Informatik Spektrum 
               31 (Springer, April 2008).
%H A016945 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A016945 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A016945 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/dimskg0new.gp">
               Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A016945 William A. Stein, <a href="http://modular.fas.harvard.edu/Tables/">The 
               modular forms database</a>
%H A016945 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CollatzProblem.html">Collatz Problem</a>
%H A016945 <a href="Sindx_3.html#3x1">Index entries for sequences related to 3x+1 
               (or Collatz) problem</a>
%F A016945 a(n) = 3(2n+1) = 3*A005408(n), odd multiples of 3.
%F A016945 a(n)=12*n-a(n-1)-12 (with a(1)=3) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 28 2009]
%e A016945 For n=2, a(2)=12*2-3-12=9; n=3, a(3)=12*3-9-12=15; n=4, a(4)=12*4-15-12=21 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 28 2009]
%p A016945 a[1]:=3:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..50); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 16 2008
%t A016945 a[n_]:=6*n+3;...and/or...Array[3+#*6&,5!,0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Dec 21 2008]
%t A016945 f[n_]:=6*n+3; lst={};Do[a=f[n];AppendTo[lst,a],{n,0,6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jun 25 2009]
%o A016945 (Other) sage: [i+3 for i in range(280) if gcd(i,6) == 6] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 20 2009]
%o A016945 (Other) sage: [crt(3, n, 3, 5) for n in xrange(3, 50)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2009]
%Y A016945 Third row of A092260.
%Y A016945 Cf. A008588, A016921, A016933, A016957, A016969.
%Y A016945 Subsequence of A061641; complement of A047263.
%Y A016945 A000225 [From L Pearson (loren.pearson(AT)gmail.com), Jul 02 2009]
%Y A016945 Sequence in context: A029506 A030594 A032676 this_sequence A110108 A162843 
               A102954
%Y A016945 Adjacent sequences: A016942 A016943 A016944 this_sequence A016946 A016947 
               A016948
%K A016945 nonn,easy,new
%O A016945 0,1
%A A016945 N. J. A. Sloane (njas(AT)research.att.com).