A126804 -id:A126804 - cp's OEIS frontend

A045943 Triangular matchstick numbers: a(n) = 3n(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

R. K. Guy

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

			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)

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

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).

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.

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

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

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

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

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 *)

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

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

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

A272244 a(n) = Product_{k=0..n} (n^2 + k^2).

Original entry on oeis.org

0, 2, 160, 21060, 4352000, 1313845000, 547573478400, 301758856490000, 212663770808320000, 186659516597629140000, 199722414913149440000000, 255947740845844788169000000, 387074162712817024892928000000, 682170272459193898736228210000000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

Cf. A126804, A272246, A272247, A272248, A367823, A367833.

Cf. A101686, A272180, A323540, A324403.

Table[Product[n^2+k^2,{k,0,n}],{n,0,15}]

a(n) ~ 2^(n + 1/2) * n^(2*(n+1)) / exp((4-Pi)*n/2).

A001814 Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m.

Original entry on oeis.org

1, 12, 180, 3360, 75600, 1995840, 60540480, 2075673600, 79394515200, 3352212864000, 154872234316800, 7771770303897600, 420970891461120000, 24481076457277440000, 1521324036987955200000, 100610229646136770560000

Offset: 1

N. J. A. Sloane

a(n) = A126804(n)/2. - Zerinvary Lajos, Sep 21 2007

a(n) is the number of ways to partition a set of 2n elements into parts of size 2 and then multiply by the number n of parts. - Alain Goupil, Jul 27 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Index entries for sequences related to Hermite polynomials

a(n) = A048854(n, 1) = A067147(2n, 2).
Cf. A001879.
Cf. A005430.

[Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

with(combinat):for n from 1 to 16 do printf(`%d, `,n!/2*sum(binomial(2*n, n), k=1..n)) od: # Zerinvary Lajos, Mar 13 2007
a:=n->sum((count(Permutation(n*2+2),size=n+1)),j=0..n)/2: seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(1/2*mul((n+k), k=1..n), n=0..16); # Zerinvary Lajos, Sep 21 2007

Table[(2*n)!/(2*(n-1)!),{n,1,20}] (* Vincenzo Librandi, Nov 22 2011 *)

combinat::catalan(n)*binomial(n+1,2)*n! $ n = 1..16; // Zerinvary Lajos, Feb 15 2007

E.g.f.: x/(1 - 4*x)^(3/2). - corrected by Alain Goupil, Jul 28 2025
a(n) = (2*n)!/(2*(n-1)!).
(n!/2)*binomial(2*n,n)*n or n!/2*A005430. - Zerinvary Lajos, Jun 06 2006
Sum_{n>=0} a(n)*x^(2n)/(2n)! = (x^2/2)*exp(x^2). - Alain Goupil, Jul 28 2025

More terms and new description from Christian G. Bower, Dec 18 2001

A272246 a(n) = Product_{k=0..n} (n^3 + k^3).

Original entry on oeis.org

0, 2, 1152, 1428840, 3488808960, 15044494500000, 105235903511101440, 1119277024472896248960, 17216259547948971039129600, 368066786222106315186876633600, 10591209807103301277597696000000000, 399472472359100444604916002033020774400

Offset: 0

Vaclav Kotesovec, Apr 23 2016

Cf. A126804, A272244, A272247, A272248, A367823, A367833.

Cf. A255433, A323541, A324426.

Table[Product[n^3+k^3,{k,0,n}],{n,0,12}]

a(n) ~ 2^(2*n + 1/2) * n^(3*n + 3) / exp((3 - Pi/sqrt(3))*n).

A272247 a(n) = Product_{k=0..n} (n^4 + k^4).

Original entry on oeis.org

0, 2, 8704, 104372388, 3087748038656, 194985808028125000, 23467500289618753093632, 4938279594477505466022892304, 1699491802948673617630927695904768, 907214902722906584628050661111614570016, 719684491044699824651608981274624000000000000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

Cf. A126804, A272244, A272246, A272248, A367823, A367833.

Cf. A255434, A323542.

Table[Product[n^4+k^4,{k,0,n}],{n,0,10}]

a(n) ~ 2^(n+1/2) * (1+sqrt(2))^(sqrt(2)*n) * n^(4*n + 4) / exp((4 - Pi/sqrt(2))*n).

A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

Original entry on oeis.org

0, 2, 67584, 7924375800, 2876035930521600, 2693451205324687500000, 5648896640332217707816550400, 23819277009290664033936067933468800, 185754160490281505676324140907107450880000, 2507604631016507710974687639612411760216253760000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

In general, for p>=1, Product_{k=0..n} (n^p + k^p) ~ sqrt(2) * n^(p*(n+1)) * exp(n*Sum_{j>=1} (-1)^(j+1) / (j*(1 + j*p))).

Cf. A126804, A272244, A272246, A272247, A367823, A367833.

Cf. A255435, A323543, A324438.

Table[Product[n^5+k^5,{k,0,n}],{n,0,10}]

a(n) ~ 2^(2*n+1/2) * phi^(sqrt(5)*n) * n^(5*n+5) / exp((5-sqrt(phi)*Pi/5^(1/4))*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A179214 Product of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

2, 6, 90, 210, 2100, 4620, 140140, 300300, 5105100, 96996900, 4481256780, 9369900540, 243617414040, 18739801080, 1164544781400, 2406725881560, 2700346439110320, 5559536786403600, 7816708721683461600

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

A073838(n) <= a(n) <= A126804(n);

a(n) = A179215(2*n)/A179215(n-1).

			a(10) = 10*11*13*14*15*17*19 = 96996900;
a(11) = 11*13*14*15*17*19*21*22 = 4481256780;
a(12) = 13*14*15*17*19*21*22*23 = 9369900540.

Cf. A179211, A179213, A005117.

a(n) = PROD(k^A008966(k): n <= k <= 2*n).

Example corrected by Reinhard Zumkeller, Jul 19 2010

A367823 a(n) = Product_{k=0..n} (n^6 + k^6).

Original entry on oeis.org

0, 2, 532480, 615291760980, 2759348178649088000, 38607000141072009765625000, 1421393314362014000255258433945600, 120987688662917113214734283836109985530000, 21538956798613753302336535534471908360548515840000, 7407676630843157974199152503244776880017496531101658740000

Offset: 0

Vaclav Kotesovec, Dec 02 2023

nonn

Cf. A126804, A272244, A272246, A272247, A272248, A324439, A367833.

Table[Product[n^6 + k^6, {k, 0, n}], {n, 0, 10}]

a(n) = prod(k=0, n, n^6 + k^6); \\ Michel Marcus, Dec 02 2023

a(n) ~ 2^(n + 1/2) * (2 + sqrt(3))^(sqrt(3)*n) * n^(6*n + 6) / exp((6 - Pi)*n).

A367833 a(n) = Product_{k=0..n} (n^8 + k^8).

Original entry on oeis.org

0, 2, 33685504, 3851231199376068, 2670340833224951538384896, 8442437541556000328575439453125000, 96982590530284083757788173242580213780447232, 3406021755102121469537208768079471859185253483110025744

Offset: 0

Vaclav Kotesovec, Dec 02 2023

nonn

Cf. A126804, A272244, A272246, A272247, A272248, A367823.

Cf. A367834.

Table[Product[n^8 + k^8, {k, 0, n}], {n, 0, 10}]

a(n) ~ 2^(n + 1/2) * ((2 + sqrt(2 + sqrt(2)))/(2 - sqrt(2 + sqrt(2))))^(sqrt(2 + sqrt(2))*n/2) * (sqrt(2) - 1 + sqrt(4 - 2*sqrt(2)))^(sqrt(2 - sqrt(2))*n) * exp((Pi/sqrt(2 - sqrt(2)) - 8)*n) * n^(8*(n + 1)).

A375845 a(n) = Product_{k=0..n} (k^8 + n).

Original entry on oeis.org

0, 2, 1548, 20400912, 2237404520000, 1316258829530177400, 3107531556500789042401392, 23981023412887138890925360910336, 519343443733819692494314622381817102336, 28055691989665530513724742545624840551562500000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

In general, for m > 2, Product_{k=0..n} (k^m + n) ~ n^(m*n + (m+1)/2) / exp(m*n - Pi * n^(1/m) / sin(Pi/m)).

Cf. A334411.

Cf. A126804 (m=1), A375839 (m=2), A375840 (m=3), A375841 (m=4), A375842 (m=5), A375843 (m=6), A375844 (m=7).

Table[Product[k^8 + n, {k, 0, n}], {n, 0, 15}]

a(n) ~ n^(8*n + 9/2) / exp(8*n - Pi*sqrt(2*(2+sqrt(2)))*n^(1/8)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

Scala

Formula

A272244 a(n) = Product_{k=0..n} (n^2 + k^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A001814 Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

MuPAD

Formula

Extensions

A272246 a(n) = Product_{k=0..n} (n^3 + k^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A272247 a(n) = Product_{k=0..n} (n^4 + k^4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A179214 Product of squarefree numbers between n and 2*n (inclusive).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A367823 a(n) = Product_{k=0..n} (n^6 + k^6).

Views

Author

Keywords

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