



0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177
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OFFSET

0,2


COMMENTS

If n != 1 and n^2+2 is prime then n is a member of this sequence.  Cino Hilliard, Mar 19 2007
Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3transversal 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
Numbers n for which polynomial 27*x^62^n is factorizable.  Artur Jasinski, Nov 01 2008
1/7 in base2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ...  Gary W. Adamson, Jan 24 2009
A165330(a(n)) = 153 for n > 0; subsequence of A031179.  Reinhard Zumkeller, Sep 17 2009
A011655(a(n)) = 0.  Reinhard Zumkeller, Nov 30 2009
A215879(a(n)) = 0.  Reinhard Zumkeller, Dec 28 2012
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
Integer areas of medial triangles of integersided triangles.
Also integer subset of A188158(n)/4.
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
From Derek Orr, Nov 22 2014: (Start)
Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n1)} + {b(0), ... b(n1)}. Then b(n+1) = a(n), for n > 0.
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)
Number of partitions of 6n into exactly 2 parts.  Colin Barker, Mar 23 2015
Partial sums are in A045943.  Guenther Schrack, May 18 2017
Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments).  Jonathan Sondow, Mar 03 2018


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107115.
Charles Cratty, Samuel Erickson, Frehiwet Negass, Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
John GrahamCumming, The hollow triangular numbers are divisible by three (2013)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315
Tanya Khovanova, Recursive Sequences
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719721.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and kcommuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 20142015.
Wikipedia, Maximal planar graphs
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

G.f.: 3*x/(1x)^2.  R. J. Mathar, Oct 23 2008
a(n) = A008486(n), n > 0.  R. J. Mathar, Oct 28 2008
a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k).  Philippe Deléham, Oct 17 2011
E.g.f.: 3*x*exp(x).  Ilya Gutkovskiy, May 18 2016
From Guenther Schrack, May 18 2017: (Start)
a(3*k) = a(a(k)) = A008591(n).
a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)


EXAMPLE

G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...


MATHEMATICA

Range[0, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)


PROG

(MAGMA) [3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011
(Maxima) makelist(3*n, n, 0, 30); /* Martin Ettl, Nov 12 2012 */
(Haskell)
a008585 = (* 3)
a008585_list = iterate (+ 3) 0  Reinhard Zumkeller, Feb 19 2013
(PARI) a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013


CROSSREFS

Row / column 3 of A004247 and of A325820.
Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products).
Cf. A031179, A008486, A008591, A017197, A017233, A045943.
Sequence in context: A209258 A296515 A031193 * A008486 A135943 A194416
Adjacent sequences: A008582 A008583 A008584 * A008586 A008587 A008588


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Partially edited by Joerg Arndt, Mar 11 2010


STATUS

approved



