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A045943

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

+10
128

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	`Also, 3 times triangular numbers, a(n) = 3A000217(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) = 3n, 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 3m(2m + 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`
	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).`
	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(3n+3) = 0. - Matthew Vandermast, Oct 28 2008` `a(n) = A000217(2n) - A000217(n-1); A179213(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010` `a(n) = a(n-1)+3n, n>0. - Vincenzo Librandi, Nov 18 2010` `G.f.: 3x/(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) = 3a(n-1)-3a(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.` `2a(n)+1 = A003215(n). - Miquel Cerda, Jan 22 2018` `a(n) = T(2n) - T(n-1), where T(n) = A000217(n). In general, T(k)T(n) = Sum_{i=0..k-1} (-1)^iT((k-i)(n-i)). - Charlie Marion, Dec 06 2020` `E.g.f.: 3exp(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(2log(2)-1)/3. (End)` `Product_{n>=1} (1 - 1/a(n)) = -(3/(2Pi))cos(sqrt(11/3)*Pi/2). - Amiram Eldar, Feb 21 2023`
	EXAMPLE	`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)`
	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 *)`
	PROG	`(Common Lisp) (defun tri (i) (if (eq i 0) 0 (+ (* 3 (- i 1)) (tri (- i 1))))) // Stephen Balaban, Jul 25 2011` `(Magma) [3n(n+1)/2: n in [0..50]]; // Vincenzo Librandi, May 02 2011` `(PARI) a(n)=3binomial(n+1, 2) \\ Charles R Greathouse IV, Jun 16 2011` `(Haskell) a n = sum [x \| x <- [n..2n]] -- Peter Kagey, Jul 27 2015` `(GAP) List([0..10^4], n -> 3n(n+1)/2); # Muniru A Asiru, Jan 24 2018` `(Scala) (3 to 150 by 3).scanLeft(0)(_ + _) // Alonso del Arte, Sep 12 2019`
	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 bn+3n(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..2n} k.` `Cf. A304993: Sum_{k = n..2n} k(k+1)/2.` `Cf. A050409: Sum_{k = n..2*n} k^2.` `Similar sequences are listed in A316466.`
	KEYWORD	`nonn,easy`
	AUTHOR	`R. K. Guy`
	STATUS	`approved`

A050409

Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

+10
13

0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Vincenzo Librandi, Table of n, a(n) for n = 0..1000` `M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013` `M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.` `Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).`
	FORMULA	`a(n) = n(n+1)(14n+1)/6.` `a(n) = A132121(n,4) for n>3. - Reinhard Zumkeller, Aug 12 2007` `From Bruno Berselli, Feb 11 2011: (Start)` `G.f.: x(5+9x)/(1-x)^4.` `a(n) = A129371(2n). (End)` `a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012` `E.g.f.: x(30 + 57x + 14x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019`
	MAPLE	`seq(add((n+k)^2, k=0..n), n=0..40); # Zerinvary Lajos, Dec 01 2006`
	MATHEMATICA	`LinearRecurrence[{4, -6, 4, -1}, {0, 5, 29, 86}, 40] (* Vincenzo Librandi, Jun 22 2012 )` `Table[(n(n+1)(14n+1))/6, {n, 0, 40}] ( Harvey P. Dale, Mar 08 2020 *)`
	PROG	`(Magma) [&+[k^2: k in [n..2n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011` `(Magma) I:=[0, 5, 29, 86]; [n le 4 select I[n] else 4Self(n-1)-6Self(n-2)+4Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012` `(PARI) a(n)=sum(k=n, n+n, k^2)` `(PARI) vector(40, n, n(n-1)(14n-13)/6) \\ G. C. Greubel, Oct 30 2019` `(Sage) [n(n+1)(14n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019` `(GAP) List([0..40], n-> n(n+1)(14*n+1)/6); # G. C. Greubel, Oct 30 2019`
	CROSSREFS	`Cf. A000330, A033994, A129371, A132112, A132121, A132124.` `Cf. A225144. - Bruno Berselli, Jun 06 2013` `Cf. A045943: Sum_{k = n..2n} k.` `Cf. A304993: Sum_{k = n..2n} k*(k+1)/2.`
	KEYWORD	`nonn,easy,nice`
	AUTHOR	`Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999`
	STATUS	`approved`

A141433

Triangle T(n, k) = (k-1)*(3*n-k), read by rows.

+10
2

0, 0, 4, 0, 7, 12, 0, 10, 18, 24, 0, 13, 24, 33, 40, 0, 16, 30, 42, 52, 60, 0, 19, 36, 51, 64, 75, 84, 0, 22, 42, 60, 76, 90, 102, 112, 0, 25, 48, 69, 88, 105, 120, 133, 144, 0, 28, 54, 78, 100, 120, 138, 154, 168, 180 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`1,3`
	LINKS	`G. C. Greubel, Rows n = 1..50 of the triangle, flattened`
	FORMULA	`G.f.: Sum_{k>=0} Sum_{n>=0} T(n,k)x^ny^k = y^2x(2x+1-3y)/((1-y)^3(x-1)^2). (G.f. for the full array, not just the triangular subspace) - R. J. Mathar, Feb 19 2020` `Sum_{k=1..n} T(n, k) = A304993(n-1) = (n-1)n(7n -2)/6. - G. C. Greubel, Apr 01 2021`
	EXAMPLE	`Triangle begins as:` `0;` `0, 4;` `0, 7, 12;` `0, 10, 18, 24;` `0, 13, 24, 33, 40;` `0, 16, 30, 42, 52, 60;` `0, 19, 36, 51, 64, 75, 84;` `0, 22, 42, 60, 76, 90, 102, 112;` `0, 25, 48, 69, 88, 105, 120, 133, 144;` `0, 28, 54, 78, 100, 120, 138, 154, 168, 180;`
	MAPLE	`A141433 := proc(n, m) (m-1)(3n-m) ; end proc:` `seq(seq(A141433(n, m), m=1..n), n=1..18) ; # R. J. Mathar, Sep 14 2011`
	MATHEMATICA	`Flatten[Table[(m-1)(3n-m), {n, 10}, {m, n}]] (* Harvey P. Dale, Feb 04 2016 *)`
	PROG	`(Magma) [(k-1)(3n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021` `(Sage) flatten([[(k-1)(3n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021`
	CROSSREFS	`Cf. A304993 (row sums).`
	KEYWORD	`nonn,easy,tabl`
	AUTHOR	`Roger L. Bagula and Gary W. Adamson, Aug 06 2008`
	STATUS	`approved`

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