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A008585 a(n) = 3*n.

%I A008585 #191 Jul 08 2025 10:46:05
%S A008585 0,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,
%T A008585 72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,
%U A008585 129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174,177
%N A008585 a(n) = 3*n.
%C A008585 If n != 1 and n^2+2 is prime then n is a member of this sequence. - _Cino Hilliard_, Mar 19 2007
%C A008585 Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3-transversal numbers): Numbers of the 3rd column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 3rd column in the square array A057145. - _Omar E. Pol_, May 02 2008
%C A008585 Numbers n for which polynomial 27*x^6-2^n is factorizable. - _Artur Jasinski_, Nov 01 2008
%C A008585 1/7 in base-2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ... - _Gary W. Adamson_, Jan 24 2009
%C A008585 A165330(a(n)) = 153 for n > 0; subsequence of A031179. - _Reinhard Zumkeller_, Sep 17 2009
%C A008585 A011655(a(n)) = 0. - _Reinhard Zumkeller_, Nov 30 2009
%C A008585 A215879(a(n)) = 0. - _Reinhard Zumkeller_, Dec 28 2012
%C A008585 Moser conjectured, and Newman proved, that the terms of this sequence are more likely to have an even number of 1s in binary than an odd number. The excess is an undulating multiple of n^(log 3/log 4). See also Coquet, who refines this result. - _Charles R Greathouse IV_, Jul 17 2013
%C A008585 Integer areas of medial triangles of integer-sided triangles.
%C A008585 Also integer subset of A188158(n)/4.
%C A008585 A medial triangle MNO is formed by joining the midpoints of the sides of a triangle ABC. The area of a medial triangle is A/4 where A is the area of the initial triangle ABC. - _Michel Lagneau_, Oct 28 2013
%C A008585 From _Derek Orr_, Nov 22 2014: (Start)
%C A008585 Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n-1)} + {b(0), ... b(n-1)}. Then b(n+1) = a(n), for n > 0.
%C A008585 Example: b(1) = the number of distinct sums of {0} + {0}. The only possible sum is {0} so b(1) = 1. b(2) = the number of distinct sums of {0,1} + {0,1}. The possible sums are {0,1,2} so b(2) = 3. b(3) = the number of distinct sums of {0,1,3} + {0,1,3}. The possible sums are {0, 1, 2, 3, 4, 6} so b(3) = 6. This continues and one can see that b(n+1) = a(n). (End)
%C A008585 Number of partitions of 6n into exactly 2 parts. - _Colin Barker_, Mar 23 2015
%C A008585 Partial sums are in A045943. - _Guenther Schrack_, May 18 2017
%C A008585 Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments). - _Jonathan Sondow_, Mar 03 2018
%C A008585 Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 3. - _Stefano Spezia_, Jul 08 2025
%D A008585 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%H A008585 Vincenzo Librandi, <a href="/A008585/b008585.txt">Table of n, a(n) for n = 0..5000</a>
%H A008585 J. Coquet, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002099551">A summation formula related to the binary digits</a>, Inventiones Mathematicae 73 (1983), pp. 107-115.
%H A008585 Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, <a href="http://www.valpo.edu/mathematics-statistics/files/2015/07/Pattern-Avoidance-in-Double-Lists.pdf">Pattern Avoidance in Double Lists</a>, preprint, 2015.
%H A008585 A. S. Fraenkel, <a href="http://www.emis.de/journals/INTEGERS/papers/eg6/eg6.Abstract.html">New games related to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
%H A008585 John Graham-Cumming, <a href="http://blog.jgc.org/2013/06/the-hollow-triangular-numbers-are.html">The hollow triangular numbers are divisible by three</a> (2013)
%H A008585 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=315">Encyclopedia of Combinatorial Structures 315</a>
%H A008585 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A008585 D. J. Newman, <a href="http://dx.doi.org/10.1090/S0002-9939-1969-0244149-8">On the number of binary digits in a multiple of three</a>, Proc. Amer. Math. Soc. 21 (1969) 719-721.
%H A008585 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H A008585 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], 2014-2015.
%H A008585 Wikipedia, <a href="https://en.wikipedia.org/wiki/Planar_graph#Maximal_planar_graphs">Maximal planar graphs</a>
%H A008585 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A008585 G.f.: 3*x/(1-x)^2. - _R. J. Mathar_, Oct 23 2008
%F A008585 a(n) = A008486(n), n > 0. - _R. J. Mathar_, Oct 28 2008
%F A008585 G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - _Gennady Eremin_, Feb 20 2021
%F A008585 a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k). - _Philippe Deléham_, Oct 17 2011
%F A008585 E.g.f.: 3*x*exp(x). - _Ilya Gutkovskiy_, May 18 2016
%F A008585 From _Guenther Schrack_, May 18 2017: (Start)
%F A008585 a(3*k) = a(a(k)) = A008591(n).
%F A008585 a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
%F A008585 a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)
%e A008585 G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...
%t A008585 Range[0, 500, 3] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%o A008585 (Magma) [3*n: n in [0..60]]; // _Vincenzo Librandi_, Jul 23 2011
%o A008585 (Maxima) makelist(3*n,n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o A008585 (Haskell)
%o A008585 a008585 = (* 3)
%o A008585 a008585_list = iterate (+ 3) 0  -- _Reinhard Zumkeller_, Feb 19 2013
%o A008585 (PARI) a(n)=3*n \\ _Charles R Greathouse IV_, Jun 28 2013
%Y A008585 Row / column 3 of A004247 and of A325820.
%Y A008585 Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).
%Y A008585 Cf. A031179, A008486, A008591, A017197, A017233, A045943.
%K A008585 nonn,easy
%O A008585 0,2
%A A008585 _N. J. A. Sloane_
%E A008585 Partially edited by _Joerg Arndt_, Mar 11 2010