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A045943 Triangular matchstick numbers: 3*n*(n+1)/2. 96
0, 3, 9, 18, 30, 45, 63, 84, 108, 135, 165, 198, 234, 273, 315, 360, 408, 459, 513, 570, 630, 693, 759, 828, 900, 975, 1053, 1134, 1218, 1305, 1395, 1488, 1584, 1683, 1785, 1890, 1998, 2109, 2223, 2340, 2460, 2583, 2709, 2838, 2970, 3105, 3243, 3384, 3528 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In the 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n = 256, ..., 511, the number of non-color partitions are computable with A045943(n-255), while for n = 512, ..., 765, the number of color points in r+g+b planes equals A000217(765-n). - Labos Elemer, Jun 20 2005

a(n) = A126890(n+1,n-1) for n>1. - Reinhard Zumkeller, Dec 30 2006

If a 3-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007

a(n) is also the smallest number that may be written both as the sum of n-1 consecutive positive integers and n consecutive positive integers. - Claudio Meller, Oct 08 2010

For n >= 3, a(n) equals 4^(2+n)*Pi^(1 - n) times the coefficient of zeta(3) in the following integral with upper bound Pi/4 and lower bound 0: int x^(n+1) tan x dx. - John M. Campbell, Jul 17 2011

Intuitively, add a triangle of three lines branching from each of the last nodes on the bottom of the graph, thus, each iteration adds 3 * (the number of nodes on the bottom of the last iteration <==> n). - Stephen Balaban, Jul 25 2011

Sequence found by reading the line from 0, in the direction 0, 3, ..., and the same line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. This is one of the orthogonal axes of the spiral; the other is A032528. - Omar E. Pol, Sep 08 2011

A005449(a(n)) = A000332(3n + 3) = C(3n + 3, 4), a second pentagonal number of triangular matchstick number index number. Additionally, a(n) - 2n is a pentagonal number (A000326). - Raphie Frank, Dec 31 2012

Sum of the numbers from n to 2n. - Wesley Ivan Hurt, Nov 24 2015

Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 5376 or 17920 or 20160. - Philippe A.J.G. Chevalier, Dec 28 2015

Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017

REFERENCES

Labos E.: On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06, 2005.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901.

Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.

Richard J. Mathar, Lozenge tilings of the equilateral triangle, Preprint 2016.

Eric Weisstein's World of Mathematics, Graph Cycle

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

Also, 3 times triangular numbers (see Crossrefs), A000217.

a(n) is the sum of n+1 integers starting from n, i.e., 1+2, 2+3+4, 3+4+5+6, 4+5+6+7+8, etc. - Jon Perry, Jan 15 2004

a(n) + A145919(3n+3) = 0. - Matthew Vandermast, Oct 28 2008

a(n) = A000217(2*n) - A000217(n-1); A179213(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010

a(n) = a(n-1)+3*n, n>0. - Vincenzo Librandi, Nov 18 2010

G.f.: 3*x/(1-x)^3. - Bruno Berselli, Jan 21 2011

a(n) = A005448(n+1) - 1. - Omar E. Pol, Oct 03 2011

a(n) = A001477(n)+A000290(n)+A000217(n). - J. M. Bergot, Dec 08 2012

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2. - Wesley Ivan Hurt, Nov 24 2015

a(n) = A027480(n)-A027480(n-1). - Peter M. Chema, Jan 18 2017.

EXAMPLE

T(n), the triangular numbers

a(n), the triangular connecting numbers

            (T(0) = 0, a(0) = 0)

       o    (T(1) = 1, a(1) = 0)

       o

      / \   (T(2) = 3, a(2) = 3)

     o - o

       o

      / \

     o - o  (T(3) = 6, a(3) = 9)

    / \ / \

   o - o - o

      ... - Stephen Balaban, Jul 25 2011

MAPLE

seq(3*binomial(n+1, 2), n=0..49); # Zerinvary Lajos, Nov 24 2006

MATHEMATICA

Table[3*n*(n + 1)/2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Oct 31 2008 *)

3 Accumulate@Range[0, 48] (* Arkadiusz Wesolowski, Oct 29 2012 *)

CoefficientList[Series[-3 x/(x - 1)^3, {x, 0, 47}], x] (* Robert G. Wilson v, Jan 29 2015 *)

LinearRecurrence[{3, -3, 1}, {0, 3, 9}, 50] (* Jean-Fran├žois Alcover, Dec 12 2016 *)

PROG

(CLISP) (defun tri (i) (if (eq i 0) 0 (+ (* 3 (- i 1)) (tri (- i 1)))))  // Stephen Balaban, Jul 25 2011

(MAGMA) [3*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, May 02 2011

(PARI) a(n)=3*binomial(n+1, 2) \\ Charles R Greathouse IV, Jun 16 2011

(Haskell) a n = sum [x | x <- [n..2*n]] -- Peter Kagey, Jul 27 2015

CROSSREFS

Cf. A005448, A002378, A046092, A051162, A126804, A001318, A032528.

3 times n-gonal numbers: A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

A diagonal of A010027.

Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A115067, A008585, A005843, A001477, A000217.

Cf. A027480 (partial sums).

Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).

Sequence in context: A100967 A193567 A134479 * A184969 A194113 A194114

Adjacent sequences:  A045940 A045941 A045942 * A045944 A045945 A045946

KEYWORD

nonn,easy

AUTHOR

R. K. Guy

STATUS

approved

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Last modified August 18 20:11 EDT 2017. Contains 290762 sequences.