A008585 a(n) = 3*n.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 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

Numbers n for which polynomial 27*x^6-2^n is factorizable. - Artur Jasinski, Nov 01 2008

1/7 in base-2 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 integer-sided 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(n-1)} + {b(0), ... b(n-1)}. 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

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

Examples

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

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. 107-115.
• Charles Cratty, Samuel Erickson, Frehiwet Negass, and 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 Graham-Cumming, 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) 719-721.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• Wikipedia, Maximal planar graphs
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Row / column 3 of A004247 and of A325820.

Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).

Cf. A031179, A008486, A008591, A017197, A017233, A045943.

Programs

Haskell

a008585 = (* 3)
a008585_list = iterate (+ 3) 0  -- Reinhard Zumkeller, Feb 19 2013

Magma

[3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Mathematica

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

Maxima

makelist(3*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013

Formula

G.f.: 3*x/(1-x)^2. - R. J. Mathar, Oct 23 2008

a(n) = A008486(n), n > 0. - R. J. Mathar, Oct 28 2008

G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - Gennady Eremin, Feb 20 2021

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)

Extensions

Partially edited by Joerg Arndt, Mar 11 2010