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%I #276 Feb 11 2024 20:44:23
%S 0,3,9,18,30,45,63,84,108,135,165,198,234,273,315,360,408,459,513,570,
%T 630,693,759,828,900,975,1053,1134,1218,1305,1395,1488,1584,1683,1785,
%U 1890,1998,2109,2223,2340,2460,2583,2709,2838,2970,3105,3243,3384,3528
%N Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.
%C Also, 3 times triangular numbers, a(n) = 3*A000217(n).
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%C Sum of the numbers from n to 2n. - _Wesley Ivan Hurt_, Nov 24 2015
%C 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
%C Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - _Eric W. Weisstein_, Jul 27 2017
%C 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
%C Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - _Bruno Berselli_, Feb 26 2018
%C Partial sums of A008585. - _Omar E. Pol_, Jun 20 2018
%C 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
%C Total number of pips in the n-th suit of a double-n domino set. - _Ivan N. Ianakiev_, Aug 23 2020
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 543.
%H Vincenzo Librandi, <a href="/A045943/b045943.txt">Table of n, a(n) for n = 0..2000</a>
%H Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, <a href="https://arxiv.org/abs/2105.00971">Enumeration of parallelogram polycubes</a>, arXiv:2105.00971 [cs.DM], 2021.
%H Francesco Brenti and Paolo Sentinelli, <a href="https://arxiv.org/abs/2212.04932">Wachs permutations, Bruhat order and weak order</a>, arXiv:2212.04932 [math.CO], 2022.
%H Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, <a href="https://arxiv.org/abs/2307.13749">Semi-simplicial combinatorics of cyclinders and subdivisions</a>, arXiv:2307.13749 [math.CO], 2023. See p. 29.
%H T. Aaron Gulliver, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/gulliverIJCMS37-40-2012.pdf">Sums of Powers of Integers Divisible by Three</a>, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1895 - 1901.
%H Alfred Hoehn, <a href="/A000326/a000326.pdf">Illustration of initial terms of A000326, A005449, A045943, A115067</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H E. Lábos, <a href="http://www.aloki.hu/pdf/0402_159169.pdf">On the number of RGB-colors we can distinguish. Partition Spectra</a>. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06, 2005. Applied Ecology and Environmental Research 4(2): 159-169, 2006.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1909.06336">Lozenge tilings of the equilateral triangle</a>, arXiv:1909.06336 [math.CO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F 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
%F a(n) = A126890(n+1,n-1) for n>1. - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) + A145919(3*n+3) = 0. - _Matthew Vandermast_, Oct 28 2008
%F a(n) = A000217(2*n) - A000217(n-1); A179213(n) <= a(n). - _Reinhard Zumkeller_, Jul 05 2010
%F a(n) = a(n-1)+3*n, n>0. - _Vincenzo Librandi_, Nov 18 2010
%F G.f.: 3*x/(1-x)^3. - _Bruno Berselli_, Jan 21 2011
%F a(n) = A005448(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = A001477(n)+A000290(n)+A000217(n). - _J. M. Bergot_, Dec 08 2012
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2. - _Wesley Ivan Hurt_, Nov 24 2015
%F a(n) = A027480(n)-A027480(n-1). - _Peter M. Chema_, Jan 18 2017.
%F 2*a(n)+1 = A003215(n). - _Miquel Cerda_, Jan 22 2018
%F 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
%F E.g.f.: 3*exp(x)*x*(2 + x)/2. - _Stefano Spezia_, May 19 2021
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2/3.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/3. (End)
%F Product_{n>=1} (1 - 1/a(n)) = -(3/(2*Pi))*cos(sqrt(11/3)*Pi/2). - _Amiram Eldar_, Feb 21 2023
%e From _Stephen Balaban_, Jul 25 2011: (Start)
%e T(n), the triangular numbers
%e a(n), the triangular connecting numbers
%e (T(0) = 0, a(0) = 0)
%e o (T(1) = 1, a(1) = 0)
%e o
%e / \ (T(2) = 3, a(2) = 3)
%e o - o
%e o
%e / \
%e o - o (T(3) = 6, a(3) = 9)
%e / \ / \
%e o - o - o
%e ... (End)
%p seq(3*binomial(n+1,2), n=0..49); # _Zerinvary Lajos_, Nov 24 2006
%t Table[3 n (n + 1)/2, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Oct 31 2008 *)
%t 3 Accumulate@Range[0, 48] (* _Arkadiusz Wesolowski_, Oct 29 2012 *)
%t CoefficientList[Series[-3 x/(x - 1)^3, {x, 0, 47}], x] (* _Robert G. Wilson v_, Jan 29 2015 *)
%t LinearRecurrence[{3, -3, 1}, {0, 3, 9}, 50] (* _Jean-François Alcover_, Dec 12 2016 *)
%o (Common Lisp) (defun tri (i) (if (eq i 0) 0 (+ (* 3 (- i 1)) (tri (- i 1))))) // _Stephen Balaban_, Jul 25 2011
%o (Magma) [3*n*(n+1)/2: n in [0..50]]; // _Vincenzo Librandi_, May 02 2011
%o (PARI) a(n)=3*binomial(n+1,2) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Haskell) a n = sum [x | x <- [n..2*n]] -- _Peter Kagey_, Jul 27 2015
%o (GAP) List([0..10^4], n -> 3*n*(n+1)/2); # _Muniru A Asiru_, Jan 24 2018
%o (Scala) (3 to 150 by 3).scanLeft(0)(_ + _) // _Alonso del Arte_, Sep 12 2019
%Y Cf. A005448, A002378, A046092, A051162, A126804, A001318, A032528.
%Y 3 times n-gonal numbers: A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
%Y 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.
%Y A diagonal of A010027.
%Y Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A115067, A008585, A005843, A001477, A000217.
%Y Cf. A027480 (partial sums).
%Y Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
%Y This sequence: Sum_{k = n..2*n} k.
%Y Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.
%Y Cf. A050409: Sum_{k = n..2*n} k^2.
%Y Similar sequences are listed in A316466.
%K nonn,easy
%O 0,2
%A _R. K. Guy_
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