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%I #119 Oct 06 2023 11:02:49
%S 3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93,99,105,111,117,123,129,
%T 135,141,147,153,159,165,171,177,183,189,195,201,207,213,219,225,231,
%U 237,243,249,255,261,267,273,279,285,291,297,303,309,315,321,327
%N a(n) = 6*n+3.
%C Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(37).
%C Continued fraction expansion of tanh(1/3).
%C 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 Janjic_, Sep 08 2007
%C 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_, Apr 17 2008
%C Let random variable X have a uniform distribution on the interval [0,c] where c is a positive constant. Then, for positive integer n, the coefficient of determination between X and X^n is (6n+3)/(n+2)^2, that is, A016945(n)/A000290(n+2). Note that the result is independent of c. For the derivation of this result, see the link in the Links section below. - _Dennis P. Walsh_, Aug 20 2013
%C Positions of 3 in A020639. - _Zak Seidov_, Apr 29 2015
%C a(n+2) gives the sum of 6 consecutive terms of A004442 starting with A004442(n). - _Wesley Ivan Hurt_, Apr 08 2016
%C Numbers k such that Fibonacci(k) mod 4 = 2. - _Bruno Berselli_, Oct 17 2017
%C Also numbers k such that t^k == -1 (mod 7), where t is a member of A047389. - _Bruno Berselli_, Dec 28 2017
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H Friedrich L. Bauer, <a href="https://doi.org/10.1007/978-3-540-85790-7_57">Der (ungerade) Collatz-Baum</a>, Informatik Spektrum 31 (Springer, April 2008), pp. 379-384.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%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 William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.
%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/POWRCORR.pdf">The correlation for a power curve on nonnegative support</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>.
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 3*(2*n + 1) = 3*A005408(n), odd multiples of 3.
%F A008615(a(n)) = n. - _Reinhard Zumkeller_, Feb 27 2008
%F A157176(a(n)) = A103333(n+1). - _Reinhard Zumkeller_, Feb 24 2009
%F a(n) = 12*n - a(n-1) for n>0, a(0)=3. - _Vincenzo Librandi_, Nov 20 2010
%F G.f.: 3*(1+x)/(1-x)^2. - _Mario C. Enriquez_, Dec 14 2016
%F E.g.f.: 3*(1 + 2*x)*exp(x). - _G. C. Greubel_, Sep 18 2019
%F Sum_{n>=0} (-1)^n/a(n) = Pi/12 (A019679). - _Amiram Eldar_, Dec 10 2021
%p seq(6*n+3, n=0..60); # _Dennis P. Walsh_, Aug 20 2013
%p A016945:=n->6*n+3; # _Wesley Ivan Hurt_, Sep 29 2013
%t Range[3, 350, 6] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t Table[6n+3, {n, 0, 60}] (* _Wesley Ivan Hurt_, Sep 29 2013 *)
%t LinearRecurrence[{2, -1}, {3, 9}, 55] (* _Ray Chandler_, Jul 17 2015 *)
%t CoefficientList[Series[3(1+x)/(1-x)^2, {x, 0, 60}], x] (* _Robert G. Wilson v_, Dec 14 2016 *)
%o (Haskell)
%o a016945 = (+ 3) . (* 6)
%o a016945_list = [3, 9 ..]
%o -- _Wesley Ivan Hurt_, Sep 29 2013
%o (Magma) [6*n+3 : n in [0..60]]; // _Wesley Ivan Hurt_, Sep 29 2013
%o (Maxima) makelist(6*n+3, n, 0, 60); /* _Wesley Ivan Hurt_, Sep 29 2013 */
%o (PARI) {a(n) = 6*n + 3} \\ _Wesley Ivan Hurt_, Sep 29 2013
%o (PARI) x='x+O('x^60); Vec(3*(1+x)/(1-x)^2) \\ _Altug Alkan_, Apr 08 2016
%o (Sage) [3*(1+2*n) for n in (0..60)] # _G. C. Greubel_, Sep 18 2019
%o (GAP) List([0..60], n-> 3*(1+2*n)); # _G. C. Greubel_, Sep 18 2019
%Y Third row of A092260.
%Y Cf. A004442, A008588, A016921, A016933, A016957, A016969, A019679.
%Y Subsequence of A061641; complement of A047263; bisection of A047241.
%Y Cf. A000225. - _Loren Pearson_, Jul 02 2009
%Y Cf. A020639. - _Zak Seidov_, Apr 29 2015
%Y Odd numbers in A355200.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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