A045943 Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.

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

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

Also, 3 times triangular numbers, a(n) = 3*A000217(n).

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

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

The difference a(n)-a(n-1) = 3*n, for n >= 1. - Stephen Balaban, Jul 25 2011 [Comment clarified by N. J. A. Sloane, Aug 01 2024]

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

Number of terms less than 10^k, k=0,1,2,3,...: 1, 3, 8, 26, 82, 258, 816, 2582, 8165, 25820, 81650, 258199, 816497, 2581989, 8164966, ... - Muniru A Asiru, Jan 24 2018

Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

Partial sums of A008585. - Omar E. Pol, Jun 20 2018

Column 1 of A273464. (Number of ways to select a unit lozenge inside an isosceles triangle of side length n; all vertices on a hexagonal lattice.) - R. J. Mathar, Jul 10 2019

Total number of pips in the n-th suit of a double-n domino set. - Ivan N. Ianakiev, Aug 23 2020

Examples

			From _Stephen Balaban_, Jul 25 2011: (Start)
T(n), the triangular numbers = number of nodes,
a(n-1) = number of edges in the T(n) graph:
       o    (T(1) = 1, a(0) = 0)
       o
      / \   (T(2) = 3, a(1) = 3)
     o - o
       o
      / \
     o - o  (T(3) = 6, a(2) = 9)
    / \ / \
   o - o - o
... [Corrected by _N. J. A. Sloane_, Aug 01 2024] (End)
		

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 543.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.
• Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
• Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 29.
• T. Aaron 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
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• E. Lábos, On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06, 2005. Applied Ecology and Environmental Research 4(2): 159-169, 2006.
• R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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).

This sequence: Sum_{k = n..2*n} k.

Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.

Cf. A050409: Sum_{k = n..2*n} k^2.

Similar sequences are listed in A316466.

Programs

GAP

List([0..10^4], n -> 3*n*(n+1)/2); # Muniru A Asiru, Jan 24 2018

Haskell

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

Magma

[3*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, May 02 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-François Alcover, Dec 12 2016 *)

PARI

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

Scala

(3 to 150 by 3).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

Formula

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

a(n) + A145919(3*n+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.

2*a(n)+1 = A003215(n). - Miquel Cerda, Jan 22 2018

a(n) = T(2*n) - T(n-1), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 06 2020

E.g.f.: 3*exp(x)*x*(2 + x)/2. - Stefano Spezia, May 19 2021

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 2/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/3. (End)

Product_{n>=1} (1 - 1/a(n)) = -(3/(2*Pi))*cos(sqrt(11/3)*Pi/2). - Amiram Eldar, Feb 21 2023