A007531 -id:A007531 - cp's OEIS frontend

A028896 6 times triangular numbers: a(n) = 3n(n+1).

0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...
The spiral begins:
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
    <==90==60==36==18===6===0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008
Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011
Partial sums of A008588. - R. J. Mathar, Aug 28 2014
Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices.  The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Leo Tavares, Illustration: Centroid Hexagons.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).
Cf. A007531.
The partial sums give A007531. - Leo Tavares, Jan 22 2022
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

List([0..44],n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019

[3*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

[seq(6*binomial(n,2),n=1..44)]; # Zerinvary Lajos, Nov 24 2006

6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *)
6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *)

a(n)=3*n*(n+1) \\ Charles R Greathouse IV, Sep 24 2015

first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018

def A028896(n): return 3*n*(n+1) # Chai Wah Wu, Aug 07 2025

O.g.f.: 6*x/(1 - x)^3.
E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017
a(n) = 6*A000217(n).
a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
From Zerinvary Lajos, Mar 06 2007: (Start)
a(n) = A049598(n)/2.
a(n) = A124080(n) - A046092(n).
a(n) = A033996(n) - A002378(n). (End)
a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008
a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011
From Philippe Deléham, Mar 26 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.
a(n) = A174709(6*n + 5). (End)
a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014
a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018
a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018
a(n) = T(3*n) - T(2*n-2) + T(n-2), 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 04 2020
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)

A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

Original entry on oeis.org

0, 0, 0, 0, 24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Also, starting with n=4, the square of area of cyclic quadrilateral with sides n, n-1, n-2, n-3. - Zak Seidov, Jun 20 2003
Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - Eric M. Schmidt, Oct 31 2012
Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
Number of 4-permutations of the set {1, 2, ..., n}. - Joerg Arndt, Apr 05 2014

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 719
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Wikipedia, Pochhammer symbol.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001094, A002378, A007531, A052787.

[n*(n-1)*(n-2)*(n-3): n in [0..30]]; // G. C. Greubel, Nov 19 2017

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,4), n=0..34); # Zerinvary Lajos, Apr 26 2007
G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 34 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..34); # Zerinvary Lajos, Apr 05 2009

Table[n*(n+1)*(n+2)*(n+3), {n,-3,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Times@@@Partition[Range[-3,60], 4, 1] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[ {5,-10,10,-5,1}, {0,0,0,0,24}, 60] (* Harvey P. Dale, May 09 2012 *)

A052762(n):=n*(n-1)*(n-2)*(n-3)$
makelist(A052762(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=24*binomial(n,4) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).
a(n) = A001094(n) - n.
E.g.f.: x^4*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.
a(n) + 1 = A062938(n-4) for n > 4. - Amarnath Murthy, Dec 13 2003
a(n) = numbperm(n, 4). - Zerinvary Lajos, Apr 26 2007
O.g.f.: -24*x^4/(-1+x)^5. - R. J. Mathar, Nov 23 2007
For n > 4: a(n) = A173333(n, n-4). - Reinhard Zumkeller, Feb 19 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24. - Harvey P. Dale, May 09 2012
a(n) = a(n-1) + 4*A007531(n). - J. M. Bergot, May 30 2012
a(n) = (n)4 = Pochhammer(n,4), using the "falling factorial" convention; other authors write Pochhammer(x,k) for what is denoted x^(k) in the Wikipedia article, then a(n) = (n-3)^(4). - _M. F. Hasler, Oct 20 2013
a(n) - 1 = A069756(n-2) for n >= 4. - Jean-Christophe Hervé, Nov 01 2015
a(n) = 24 * A000332(n). - Bruce J. Nicholson, Apr 03 2017
From R. J. Mathar, Jun 30 2021: (Start)
Sum_{n>=4} 24*(-1)^n/a(n) = A242023.
Sum_{n>=4} 1/a(n) = 1/18. (End)

More terms from Henry Bottomley, Mar 20 2000

Formula corrected by Philippe Deléham, Dec 12 2003

A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Reinhard Zumkeller, Feb 19 2010

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Index entries for sequences related to factorial numbers.

Row sums give A002627.
Central terms give A006963:
T(2*n-1,n) = A006963(n+1).
T(2*n,n) = A001813(n).
T(2*n,n+1) = A001761(n).
1 < k <= n: T(n,k) = T(n,k-1) / k.
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).
1 <= k <= n: T(n+1,k+1) = A162995(n,k).
T(n,1) = A000142(n).
T(n,2) = A001710(n) for n>1.
T(n,3) = A001715(n) for n>2.
T(n,4) = A001720(n) for n>3.
T(n,5) = A001725(n) for n>4.
T(n,6) = A001730(n) for n>5.
T(n,7) = A049388(n-7) for n>6.
T(n,8) = A049389(n-8) for n>7.
T(n,9) = A049398(n-9) for n>8.
T(n,10) = A051431(n) for n>9.
T(n,n-7) = A159083(n+1) for n>7.
T(n,n-6) = A053625(n+1) for n>6.
T(n,n-5) = A052787(n) for n>5.
T(n,n-4) = A052762(n) for n>4.
T(n,n-3) = A007531(n) for n>3.
T(n,n-2) = A002378(n-1) for n>2.
T(n,n-1) = A000027(n) for n>1.
T(n,n) = A000012(n).
Cf. A138533, A002627.

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

A007675 Numbers m such that m, m+1 and m+2 are squarefree.

Original entry on oeis.org

1, 5, 13, 21, 29, 33, 37, 41, 57, 65, 69, 77, 85, 93, 101, 105, 109, 113, 129, 137, 141, 157, 165, 177, 181, 185, 193, 201, 209, 213, 217, 221, 229, 237, 253, 257, 265, 281, 285, 301, 309, 317, 321, 329, 345, 353, 357, 365, 381, 389, 393, 397, 401, 409, 417, 429, 433, 437, 445, 453

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Four categories: all terms are composites like {33, 34, 35}; first term only is prime like {37, 38, 39}; third term only is prime like {57, 58, 59}; first and third are primes like {29, 30, 31}. - Labos Elemer
Four consecutive integers cannot be squarefree as one of them is divisible by 2^2 = 4. - Amarnath Murthy, Feb 18 2002
Numbers m such that m^3 + 3m^2 + 2m is squarefree. See proof below. - Charles R Greathouse IV, Mar 05 2013
There are kx + O(x/log x) terms of this sequence below x, where k = A206256. - Charles R Greathouse IV, Mar 05 2013
Proof: m^3 + 3m^2 + 2m = m*(m+1)*(m+2) and the factors are pairwise relatively prime, because (m+1) is even. - Thomas Ordowski, Apr 20 2013
Conjecture: for every prime p, the numbers p# - 1, p#, p# + 1 are squarefree, where primorial p# = product of all primes <= p. - Thomas Ordowski, Apr 21 2013
Let f(m) = abs(mu(m*(m+1)*(m+2))), where mu(m) is the Moebius function, then the sum S(m) = f(1) + f(2) + ... + f(m) ~ k*m with the constant k = A206256 = 0.12548698.... - Thomas Ordowski, Apr 22 2013
All terms are congruent to 1 (mod 4). - Zak Seidov, Dec 22 2014

			85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.

P. R. Halmos, Problems for Mathematicians Young and Old. Math. Assoc. America, 1991, p. 28.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Subsequence of A007674, A016813, and A056911.

Cf. A005117, A013929, A008966, A007531.

a007675 n = a007675_list !! (n-1)
a007675_list = f 1 a008966_list where
   f n (u:xs'@(v:w:x:xs)) | u == 1 && w == 1 && v == 1 = n : f (n+4) xs
                          | otherwise = f (n+1) xs'
-- Reinhard Zumkeller, Nov 05 2011

select(t -> andmap(NumberTheory:-IsSquareFree,[t,t+1,t+2]), [seq(i,i=1..1000,4)]); # Robert Israel, Jul 16 2024

Select[Range[1000], SquareFreeQ[#(# + 1)(# + 2)] &] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Transpose[Select[Partition[Select[Range[400], SquareFreeQ], 3, 1], Differences[#] == {1, 1} &]][[1]] (* Harvey P. Dale, Apr 11 2012 *)
Select[Range[1, 499, 2], MoebiusMu[#^3 + 3#^2 + 2#] != 0 &] (* Alonso del Arte, Jan 16 2014 *)
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,500}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 14 2017 *)

is(n)=issquarefree(n)&&issquarefree(n+1)&&issquarefree(n+2) \\ Charles R Greathouse IV, Mar 05 2013

Numbers m such that g(m)*g(m+1)*g(m+2) = 1, where g(w) = abs(mu(w)). - Labos Elemer

a(n) ~ c*n with c = 7.96895... = 1/A206256. - Charles R Greathouse IV, Mar 05 2013

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

Original entry on oeis.org

0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, 278256, 314160, 353430, 396270, 442890, 493506, 548340, 607620

Offset: 0

N. J. A. Sloane

Non-vanishing diagonal of (A132440)^4/4. Third subdiagonal of unsigned A238363 without the zero. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of the complete graph K_4. - Tom Copeland, Apr 05 2014
Total number of pips on a set of trominoes (3-armed dominoes) with up to n pips on each arm. - Alan Shore and N. J. A. Sloane, Jan 06 2016
Also the number of minimum connected dominating sets in the (n+2)-crown graph. - Eric W. Weisstein, Jun 29 2017
Crossing number of the (n+3)-cocktail party graph (conjectured). - Eric W. Weisstein, Apr 29 2019
Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - Edward Krogius, Jul 31 2022

			G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Vincenzo Librandi, Table of n, a(n) for n = 0..690
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Aleksandar Petojević and Nenad Đapić, The vAm(a,b,c;z) function, Preprint 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Hasan Unal, Proof Without Words: Sums of Products of Three Consecutive Integers, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Crown Party Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Index entries for sequences related to Bessel functions or polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A002378, A033486, A033488, A034827, A050534, A130534, A132440, A238363.
Partial sums of A007531.
A row of the array in A129533.
A column of the triangle in A331430.
Sequences of the form binomial(n+k,k)*binomial(n+k+2,k): A000012 (k=0), A005563 (k=1), this sequence (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).

[n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(binomial(n+3,4)*6, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* Harvey P. Dale, Nov 27 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* Harvey P. Dale, Nov 27 2013 *)
Table[6 Binomial[n+3, 4], {n,0,40}] (* Eric W. Weisstein, Jun 29 2017 *)
Times @@@ Table[n+k, {n, 0, 40}, {k, 0, 3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n)=6*binomial(n+3,4) \\ Charles R Greathouse IV, Apr 17 2012

concat(0, Vec(6*x/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 29 2015

def A033487(n): return 6*binomial(n+3,4)
print([A033487(n) for n in range(41)]) # G. C. Greubel, Feb 08 2025

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)
G.f.: 6*x/(1-x)^5.
a(n) = 6*binomial(n+3, 4) = 6*A000332(n+3).
a(n) = a(n-1) + A007531(n+1).
a(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)
Constant term in Bessel polynomial {y_n(x)}''.
a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - Zerinvary Lajos, May 25 2005
a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - Zerinvary Lajos, May 17 2006
From Zerinvary Lajos, May 11 2007: (Start)
a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j.
a(n) = Sum_{j=1..n} j*(n+2)*(n-1)/2. (End)
Sum_{n>0} 1/a(n) = 2/9. - Enrique Pérez Herrero, Nov 10 2013
a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - Michael Somos, Apr 06 2014
a(n) = A002378(binomial(n+2,2)-1). - Salvador Cerdá, Nov 04 2016
a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - Michel Marcus, Oct 29 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/4. - Stefano Spezia, Jul 03 2025

A047217 Numbers that are congruent to {0, 1, 2} mod 5.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47, 50, 51, 52, 55, 56, 57, 60, 61, 62, 65, 66, 67, 70, 71, 72, 75, 76, 77, 80, 81, 82, 85, 86, 87, 90, 91, 92, 95, 96, 97, 100, 101, 102, 105, 106, 107, 110, 111

Offset: 1

N. J. A. Sloane

Also, the only numbers that are eligible to be the sum of two 4th powers (A004831). - Cino Hilliard, Nov 23 2003
Nonnegative m such that floor(2*m/5) = 2*floor(m/5). - Bruno Berselli, Dec 09 2015
The sequence lists the indices of the multiples of 5 in A007531. - Bruno Berselli, Jan 05 2018

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A007531, A030341, A004831 (two 4th powers).

Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.

I:=[0, 1, 2, 5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 25 2012

&cat [[5*n,5*n+1,5*n+2]: n in [0..30]]; // Bruno Berselli, Dec 09 2015

seq(op([5*i,5*i+1,5*i+2]),i=0..100); # Robert Israel, Sep 02 2014

Select[Range[0,120], MemberQ[{0,1,2}, Mod[#,5]]&] (* Harvey P. Dale, Jan 20 2012 *)

a(n)=n--\3*5+n%3 \\ Charles R Greathouse IV, Oct 22 2011

concat(0, Vec(x^2*(1+x+3*x^2)/(1-x)^2/(1+x+x^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015

is(n) = n%5 < 3 \\ Felix Fröhlich, Jan 05 2018

a(n+1) = Sum_{k>=0} A030341(n,k)*b(k) with b(0)=1 and b(k)=5*3^(k-1) for k>0. - Philippe Deléham, Oct 22 2011
G.f.: x^2*(1+x+3*x^2)/(1-x)^2/(1+x+x^2). - Colin Barker, Feb 17 2012
a(n) = 5 + a(n-3) for n>3. - Robert Israel, Sep 02 2014
a(n) = floor((5/4)*floor(4*(n-1)/3)). - Bruno Berselli, May 03 2016
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3*k) = 5*k-3, a(3*k-1) = 5*k-4, a(3*k-2) = 5*k-5. (End)
a(n) = n - 1 + 2*floor((n-1)/3). - Bruno Berselli, Feb 06 2017
Sum_{n>=2} (-1)^n/a(n) = sqrt(1-2/sqrt(5))*Pi/5 + 3*log(2)/5. - Amiram Eldar, Dec 10 2021

A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120, 65280, 19656, 990, 0

Offset: 1

Alois P. Heinz, Aug 07 2008

Column k is Dirichlet convolution of mu(n) with k^n.

The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.

			T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,  2,   3,    4,    5, ...
  0,  2,   6,   12,   20, ...
  0,  6,  24,   60,  120, ...
  0, 12,  72,  240,  600, ...
  0, 30, 240, 1020, 3120, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
C. J. Smyth, A coloring proof of a generalisation of Fermat's little theorem, Amer. Math. Monthly 93, No. 6 (1986), pp. 469-471.
Lucas Vieira Barbosa, Nonclassical Temporal Correlations Under Finite-Memory Constraints, Doctoral Thesis, Univ. Wien (Austria 2024). See p. 35.
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127.
Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133.
Main diagonal gives A252764.
Cf. A074650, A143325, A008683, A054525.

with(numtheory): f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T:= (n,k)-> f0(n)(k); seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = Sum_{d|n} k^d * mu(n/d).

T(n,k) = k^n - Sum_{d

T(n,k) = A143325(n,k) * k.

T(n,k) = A074650(n,k) * n.

So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021

A011886 a(n) = floor(n(n-1)(n-2)/4).

Original entry on oeis.org

0, 0, 0, 1, 6, 15, 30, 52, 84, 126, 180, 247, 330, 429, 546, 682, 840, 1020, 1224, 1453, 1710, 1995, 2310, 2656, 3036, 3450, 3900, 4387, 4914, 5481, 6090, 6742, 7440, 8184, 8976, 9817, 10710, 11655, 12654, 13708, 14820, 15990, 17220, 18511, 19866, 21285

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Sequences of the form floor(n*(n-1)*(n-2)/m): A007531 (m=1), A135503 (m=2), A007290 (m=3), this sequence (m=4), A011887 (m=5), A000292 (m=6), A011889 (m=7), A011890 (m=8), A011891 (m=9), A011892 (m=10), A011893 (m=11), A011894 (m=12), A011895 (m=13), A011896 (m=14), A011897 (m=15), A011898 (m=16), A011899 (m=17), A011849 (m=18), A011901 (m=19), A011902 (m=20), A011903 (m=21), A011904 (m=22), A011905 (m=23), A011842 (m=24), A011907 (m=25), A011908 (m=26), A011909 (m=27), A011910 (m=28), A011911 (m=29), A011912 (m=30), A011912 (m=31), A011913 (m=32).

[Floor(n*(n-1)*(n-2)/4): n in [0..50]]; // Vincenzo Librandi, Jul 07 2012

Table[Floor[(n(n-1)(n-2))/4],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1,1, -3,3,-1},{0,0,0,1,6,15,30}, 50] (* Harvey P. Dale, Feb 25 2012 *)
CoefficientList[Series[x^3*(1+3*x+2*x^3)/((1-x)^3*(1-x^4)),{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)

[3*binomial(n,3)//2 for n in range(51)] # G. C. Greubel, Oct 06 2024

From R. J. Mathar, Apr 15 2010: (Start)
a(n) = +3*a(n-1) -3*a(n-2) +a(n-3) +a(n-4) -3*a(n-5) +3*a(n-6) -a(n-7).
G.f.: x^3*(1+3*x+2*x^3) / ( (1-x)^4*(1+x)*(1+x^2) ). (End)
a(n) = floor(Sum_{k=0..n} n*(k+1)/2) for n >= -2. - William A. Tedeschi, Sep 10 2010

More terms from William A. Tedeschi, Sep 10 2010

A014125 Bisection of A001400.

Original entry on oeis.org

1, 3, 6, 11, 18, 27, 39, 54, 72, 94, 120, 150, 185, 225, 270, 321, 378, 441, 511, 588, 672, 764, 864, 972, 1089, 1215, 1350, 1495, 1650, 1815, 1991, 2178, 2376, 2586, 2808, 3042, 3289, 3549, 3822, 4109, 4410, 4725, 5055, 5400, 5760, 6136, 6528, 6936, 7361, 7803

Offset: 0

N. J. A. Sloane

Also Schoenheim bound L_1(n,5,4).
Degrees of polynomials defined by p(n) = (x^(n+1)*p(n-1)p(n-3) + p(n-2)^2)/p(n-4), p(-4)=p(-3)=p(-2)=p(-1)=1. - Michael Somos, Jul 21 2004
Degrees of polynomial tau-functions of q-discrete Painlevé I, which generate sequence A095708 when q=2 (up to an offset of 3). - Andrew Hone, Jul 29 2004
Because of the Laurent phenomenon for the general q-discrete Painlevé I tau-function recurrence p(n) = (a*x^(n+1)*p(n-1)*p(n-3) + b*p(n-2)^2)/p(n-4), p(n) for n > -1 will always be a polynomial in x and a Laurent polynomial in a,b and the initial data p(-4),p(-3),p(-2),p(-1). - Andrew Hone, Jul 29 2004
Create the sequence 0,0,0,0,0,6,18,36,66,108,... so that the sum of three consecutive terms b(n) + b(n+1) + b(n+2) = A007531(n), with b(0)=0; then a(n) = b(n+5)/6. - J. M. Bergot, Jul 30 2013
Number of partitions of n into three kinds of part 1 and one kind of part 3. - Joerg Arndt, Sep 28 2015
First differences are A001840(k) starting with k=2; second differences are A086161(k) starting with k=1. - Bob Selcoe, Sep 28 2015
Maximum Wiener index of all maximal planar graphs with n+2 vertices.  The extremal graphs are cubes of paths. - Allan Bickle, Jul 09 2022
Maximum Wiener index of all maximal 3-degenerate graphs with n+2 vertices.  (A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices.)  The extremal graphs are cubes of paths, so the bound also applies to 3-trees. - Allan Bickle, Sep 18 2022

			Polynomials: p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...
a(12)=185:  A000217(13)=91 + a(9)=94 == 91+55+28+10+1 = 185. - _Bob Selcoe_, Sep 27 2015
a(3)=11: the 11 partitions of 3 are {1a,1a,1a}, {1a,1a,1b}, {1a,1a,1c}, {1a,1b,1b}, {1a,1b,1c}, {1a,1c,1c}, {1b,1b,1b}, {1b,1b,1c}, {1b,1c,1c}, {1c,1c,1c}, {3}. - _Bob Selcoe_, Oct 04 2015

W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992. See Eq. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
L. Smiley, Hidden Hexagons (preprint).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Joshua Alman, Cesar Cuenca, and Jiaoyang Huang, Laurent phenomenon sequences, Journal of Algebraic Combinatorics 43(3) (2015), 589-633.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Z. Che and K. L. Collins, An upper bound on Wiener indices of maximal planar graphs, Discrete Appl. Math. 258 (2019), 76-86.
Éva Czabarka, Peter Dankelmann, Trevor Olsen, and László A. Székely, Wiener Index and Remoteness in Triangulations and Quadrangulations, arXiv:1905.06753 [math.CO], 2019.
S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.
D. Ghosh, E. Győri, A. Paulos, N. Salia, and O. Zamora, The maximum Wiener index of maximal planar graphs, Journal of Combinatorial Optimization 40, (2020), 1121-1135.
H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=3]
Index entries for two-way infinite sequences
Index entries for covering numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A000217, A000631, A001840, A014126, A086161, A095708.
A column of A036838.
Cf. A002623, A014125, A122046, A122047. - Johannes W. Meijer, May 20 2011
Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, A175724.

[n^3/18+n^2/2+4*n/3+1+(((n+1) mod 3)-1)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 14 2015

I:=[1,3,6,11,18,27]; [n le 6 select I[n] else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3)-3*Self(n-4)+3*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Apr 15 2015

L := proc(v,k,t,l) local i,t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v,k,t). Current sequence is L_1(n,n-3,n-4,1).

CoefficientList[Series[1/((1 - x)^3*(1 - x^3)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 14 2015 *)

a(n)=if(n<-5,-a(-6-n),polcoeff(1/(1-x)^3/(1-x^3)+x^n*O(x),n)) /* Michael Somos, Jul 21 2004 */

my(x='x+O('x^50)); Vec(1/((1-x)^3*(1-x^3))) \\ Altug Alkan, Oct 16 2015

a(n)=(n^3 + 9*n^2 + 24*n + 19)\/18 \\ Charles R Greathouse IV, Jun 29 2020

[(binomial(n+4,3) - ((n+4)//3))/3 for n in (0..50)] # G. C. Greubel, Apr 28 2019

G.f.: 1/((1-x)^3*(1-x^3)).
a(n) = -a(-6-n) = 3*a(n-1) -3*a(n-2) +2*a(n-3) -3*a(n-4) +3*a(n-5) -a(n-6).
The simplest recurrence is fourth order: a(n) = a(n-1) + a(n-3) - a(n-4) + n + 1, which gives the g.f.: 1/((1-x)^3*(1-x^3)), with a(-4) = a(-3) = a(-2) = a(-1) = 0.
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + 2/(9*sqrt(3))*sin(2*Pi*n/3). - Andrew Hone, Jul 29 2004
a(n) = n^3/18 + n^2/2 + 4*n/3 + 1 + (((n+1) mod 3) - 1)/9. - same formula, simplified by Gerald Hillier, Apr 14 2015
a(n) = (2*A000027(n+1) + 3*A000292(n+1) + A049347(n-1) + 1 + 3*A000217(n+1))/9. - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A144677(n) + A144677(n-1) + A144677(n-2).
a(n) = A190717(n-4) + 2*A190717(n-3) + 3*A190717(n-2) + 2*A190717(n-1) + A190717(n). (End)
3*a(n) = binomial(n+4,3) - floor((n+4)/3). - Bruno Berselli, Nov 08 2013
a(n) = A000217(n+1) + a(n-3) = Sum_{j>=0, n>=3*j} (n-3*j+1)*(n-3*j+2)/2. - Bob Selcoe, Sep 27 2015
a(n) = round(((2*n+5)^3 + 3*(2*n+5)^2 - 9*(2*n+5))/144). - Giacomo Guglieri, Jun 28 2020
a(n) = floor(((n+2)^3 + 3*(n+2)^2)/18). - Allan Bickle, Aug 01 2020
a(n) = Sum_{j=0..n} (n-j+1)*floor((j+3)/3). - G. C. Greubel, Oct 18 2021
E.g.f.: exp(x) + exp(x)*x*(34 + 12*x + x^2)/18 + 2*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - Stefano Spezia, Apr 05 2023

More terms from James Sellers, Dec 24 1999

A143325 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1) which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 6, 0, 1, 4, 15, 24, 15, 0, 1, 5, 24, 60, 80, 27, 0, 1, 6, 35, 120, 255, 232, 63, 0, 1, 7, 48, 210, 624, 1005, 728, 120, 0, 1, 8, 63, 336, 1295, 3096, 4095, 2160, 252, 0, 1, 9, 80, 504, 2400, 7735, 15624, 16320, 6552, 495, 0, 1, 10, 99

Offset: 1

Alois P. Heinz, Aug 07 2008

Column k is Dirichlet convolution of mu(n) with k^(n-1). The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k+k^3.

			T(4,2)=6, because 6 words of length 4 over 2-letter alphabet {a,b} are primitive and earlier than others derived by cyclic shifts of the alphabet: aaab, aaba, aabb, abaa, abba, abbb; note that aaaa and abab are not primitive and words beginning with b can be derived by shifts of the alphabet from words in the list; secondly note that the words in the list need not be Lyndon words, for example aaba can be derived from aaab by a cyclic rotation of the positions.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  0,   1,    2,     3,     4,      5,      6,       7, ...
  0,   3,    8,    15,    24,     35,     48,      63, ...
  0,   6,   24,    60,   120,    210,    336,     504, ...
  0,  15,   80,   255,   624,   1295,   2400,    4095, ...
  0,  27,  232,  1005,  3096,   7735,  16752,   32697, ...
  0,  63,  728,  4095, 15624,  46655, 117648,  262143, ...
  0, 120, 2160, 16320, 78000, 279720, 823200, 2096640, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A063524, A000740, A034741, A295505, A295506, A320071, A320072, A320073, A320074, A320075.
Rows n=1-5, 7 give: A000012, A001477, A005563, A007531, A123865, A123866.
Main diagonal gives A075147.
Cf. A074650, A143324, A008683, A054525.

with(numtheory):
f1:= proc(n) option remember;
       unapply(k^(n-1)-add(f1(d)(k), d=divisors(n)minus{n}), k)
     end;
T:= (n,k)-> f1(n)(k);
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 21 2014, from first formula *)

T(n,k) = Sum_{d|n} k^(d-1) * mu(n/d).

T(n,k) = k^(n-1) - Sum_{d

T(n,k) = A074650(n,k) * n/k.

T(n,k) = A143324(n,k) / k.

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A028896 6 times triangular numbers: a(n) = 3*n*(n+1).

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A052762 Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).

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A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

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A007675 Numbers m such that m, m+1 and m+2 are squarefree.

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A033487 a(n) = n*(n+1)*(n+2)*(n+3)/4.

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A047217 Numbers that are congruent to {0, 1, 2} mod 5.

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A028896 6 times triangular numbers: a(n) = 3n(n+1).

A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

A011886 a(n) = floor(n(n-1)(n-2)/4).