A007531 -id:A007531 - cp's OEIS frontend

A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 24, 5, 0, 0, 6, 192, 60, 6, 0, 0, 6, 2112, 1620, 120, 7, 0, 0, 6, 32640, 98820, 7680, 210, 8, 0, 0, 6, 718080, 13638780, 1574400, 26250, 336, 9, 0, 0, 6, 22665216, 4260983940, 1034019840, 13676250, 72576, 504, 10

Offset: 1

Alois P. Heinz, Dec 02 2010

The k X k X k triangular grid has k rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(k) vertices and 3*A000217(k-1) edges altogether.

The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - Georg Fischer, Jul 31 2023

			Square array A(n,k) begins:
  1,   0,    0,       0,          0,             0,  ...
  2,   0,    0,       0,          0,             0,  ...
  3,   6,    6,       6,          6,             6,  ...
  4,  24,  192,    2112,      32640,        718080,  ...
  5,  60, 1620,   98820,   13638780,    4260983940,  ...
  6, 120, 7680, 1574400, 1034019840, 2175789895680,  ...

Wikipedia, Triangular grid graph
Wikipedia, Chromatic polynomial

Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
Rows n=1-10 give: A000007(k-1), A000038(k-1), A040006(k-1), A182798, A153467*4, A153468*5, A153469*6, A153470*7, A153471*8, A153472*9, A153473*10.
Cf. A000217, A193283.

A045619 Numbers that are the products of 2 or more consecutive integers.

Original entry on oeis.org

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680

Offset: 1

Erich Friedman

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597). - T. D. Noe, Oct 13 2002

Numbers of the form x!/y! with y+1 < x. - Reinhard Zumkeller, Feb 20 2008

			30 is in the sequence as 30 = 5*6 = 5*(5+1). - _David A. Corneth_, Oct 19 2021

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T.D. Noe)
P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292-301.

Union of A002378, A007531, A052762, A052787, A053625, etc. - R. J. Mathar, Oct 19 2021

Cf. A001597, A000142, A137895, A093449, A064224, A084720, A137899, A137900, A120436, A097889.

maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]

list(lim)=my(v=List([0]),P,k=1,t); while(1, k++; P=binomial('n+k-1,k)*k!; if(subst(P,'n,1)>lim, break); for(n=1,lim, t=eval(P); if(t>lim, next(2)); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Nov 16 2021

import heapq
from sympy import sieve
def aupton(terms, verbose=False):
    p = 6; h = [(p, 2, 3)]; nextcount = 4; aset = {0, 2}
    while len(aset) < terms:
        (v, s, l) = heapq.heappop(h)
        aset.add(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i]")
        if v >= p:
            p *= nextcount
            heapq.heappush(h, (p, 2, nextcount))
            nextcount += 1
        v //= s; s += 1; l += 1; v *= l
        heapq.heappush(h, (v, s, l))
    return sorted(aset)
print(aupton(52)) # Michael S. Branicky, Oct 19 2021

a(n) = A000142(A137911(n))/A000142(A137912(n)-1) for n>1. - Reinhard Zumkeller, Feb 27 2008
Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n-1) n ~ n^2. - Daniel Forgues, Mar 26 2012
a(n) = n^2 - 2*n^(5/3) + O(n^(4/3)). - Charles R Greathouse IV, Aug 27 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
More terms from Reinhard Zumkeller, Feb 27 2008
Incorrect program removed by David A. Corneth, Oct 19 2021

A052787 Product of 5 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 720, 2520, 6720, 15120, 30240, 55440, 95040, 154440, 240240, 360360, 524160, 742560, 1028160, 1395360, 1860480, 2441880, 3160080, 4037880, 5100480, 6375600, 7893600, 9687600, 11793600, 14250600, 17100720, 20389320, 24165120, 28480320, 33390720

Offset: 0

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

Appears in Harriot along with the formula (for a different offset) a(n) = n^5 + 10n^4 + 35n^3 + 50n^2 + 24n, see links. - Charles R Greathouse IV, Oct 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Thomas Harriot, Manuscript 6782, p. 77, c. 1599.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 744.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A052762, A000389, A054559, A173333.

[n*(n-1)*(n-2)*(n-3)*(n-4): n in [0..35]]; // Vincenzo Librandi, May 26 2011

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,5), n=0..31); # Zerinvary Lajos, Apr 26 2007
G(x):=x^5*exp(x): f[0]:=G(x): for n from 1 to 31 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..31); # Zerinvary Lajos, Apr 05 2009

Times@@@(Partition[Range[-4,35],5,1])  (* Harvey P. Dale, Feb 04 2011 *)

a(n)=120*binomial(n,5) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)=n!/(n-5)!. [Corrected by Philippe Deléham, Dec 12 2003]
a(n) = 120*A000389(n) = 4*A054559(n).
E.g.f.: x^5*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-1-n)*a(n)+(-4+n)*a(n+1), a(5)=120}.
O.g.f.: 120*x^5/(-1+x)^6. - R. J. Mathar, Nov 16 2007
For n>5: a(n) = A173333(n,n-5). - Reinhard Zumkeller, Feb 19 2010
a(n) = a(n-1) + 5*A052762(n). - J. M. Bergot, May 30 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/96.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*log(2)/3 - 131/288. (End)

More terms from Henry Bottomley, Mar 20 2000

A055112 a(n) = n(n+1)(2*n+1).

Original entry on oeis.org

0, 6, 30, 84, 180, 330, 546, 840, 1224, 1710, 2310, 3036, 3900, 4914, 6090, 7440, 8976, 10710, 12654, 14820, 17220, 19866, 22770, 25944, 29400, 33150, 37206, 41580, 46284, 51330, 56730, 62496, 68640, 75174, 82110, 89460, 97236, 105450

Offset: 0

Henry Bottomley, Jun 15 2000

Original name: Areas of Pythagorean triangles (X, Y, Z = Y + 1) with X^2 + Y^2 = Z^2.
a(n) is the set of possible y values for 4*x^3 + x^2 = y^2 with the x values being A002378(n). - Gary Detlefs, Feb 22 2010
This sequence is related to A028896 by a(n) = n*A028896(n) - Sum_{i = 0..n-1} A028896(i) and this is the case d = 3 in the identity n*(d*(d+1)*n*(n+1)/4) - Sum_{i = 0..n-1} d*(d+1)*i*(i+1)/4 = d*(d+1)*n*(n+1)*(2*n+1)/12. - Bruno Berselli, Mar 31 2012
Also sums of rows of natural numbers (cf. A001477) seen as triangle with an odd numbers of terms per row, see example. - Reinhard Zumkeller, Jan 24 2013
Without mentioning the connection to Pythagorean triangles, Bolker (1967) gives it as an exercise to prove that these numbers are always divisible by 6. This is easy to prove from the formula that he gives, n(n - 1)(2n - 1): obviously either n or (n - 1) must be even; then, if n is congruent to 2 mod 3 it means that (2n - 1) is a multiple of 3, otherwise either n or (n - 1) is a multiple of 3; thus both prime divisors of 6 are accounted for in a(n). - Alonso del Arte, Oct 13 2013
a(n) = n*(n+1)*(n+(n+1)) is the product of two consecutive integers multiplied by the sum of those two consecutive integers. - Charles Kusniec, Sep 04 2022

			.  n   A001477(n) as triangle with row lengths = 2*n+1   Row sums = a(n)
.  0                         0                                  0
.  1                      1  2  3                               6
.  2                   4  5  6  7  8                           30
.  3                9 10 11 12 13 14 15                        84
.  4            16 17 18 19 20 21 22 23 24                    180
.  5         25 26 27 28 29 30 31 32 33 34 35                 330
.  6      36 37 38 39 40 41 42 43 44 45 46 47 48              546
.  7   49 50 51 52 53 54 55 56 57 58 59 60 61 62 63           840 .
- _Reinhard Zumkeller_, Jan 24 2013

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Mathematics Magazine, 63 (5) 1990, 291-306.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408 (X values), A046092 (Y values), A001844 (Z values), A002939 (perimeter), A033581.
Similar sequences are listed in A316224.
Cf. A000217, A000290, A005898, A083374.

Table[n(n + 1)(2n + 1), {n, 0, 39}] (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,6,30,84},50] (* Harvey P. Dale, Oct 02 2024 *)

PARI
```
a(n)=n*(n+1)*(2*n+1);
```

def A055112(n): return n*(n*((n<<1) + 3) + 1) # Chai Wah Wu, Nov 14 2022

a(n) = n*(n+1)*(2*n+1).
G.f.: 6*x*(1+x)/(1-x)^4. - Bruno Berselli, Mar 31 2012
From Benoit Cloitre, Apr 30 2002: (Start)
a(n) = 6*A000330(n) = A007531(2*n)/4 = 3*A000292(2*n-1)/2 = A005408(n)*A046092(n)/2 = A005408(n)*(A001844(n)-1)/2.
Sum_{n > 0} 1/a(n) = 3 - 4*log(2). (End)
a(n) = Sum_{i = 1..n} A033581(i). - Jonathan Vos Post, Mar 15 2006
a(n) = A000217(2*n)*A000217(2*n+1)/(2*n+1). - Charlie Marion, Feb 17 2012
a(n) = Sum_{i = 1..2*n + 1} (n^2 + (i-1)). - Charlie Marion, Sep 14 2012
Sum_{n >= 1} (-1)^(n+1)/a(n) = Pi - 3, due to Nilakantha, circa 1500. See Roy p. 304. - Peter Bala, Feb 19 2015
a(n) = A002378(n) * (2n+1). - Bruce J. Nicholson, Aug 31 2017
a(n) = Sum_{k=n^2..(n+1)^2-1} k. - Darío Clavijo, Jan 31 2025
E.g.f.: exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Feb 02 2025
a(n) = A005898(n) - A005408(n) = A083374(n+1) - A083374(n). - J.S. Seneschal, Jul 08 2025

A054602 a(n) = Sum_{d|3} phi(d)*n^(3/d).

Original entry on oeis.org

0, 3, 12, 33, 72, 135, 228, 357, 528, 747, 1020, 1353, 1752, 2223, 2772, 3405, 4128, 4947, 5868, 6897, 8040, 9303, 10692, 12213, 13872, 15675, 17628, 19737, 22008, 24447, 27060, 29853, 32832, 36003, 39372, 42945, 46728, 50727, 54948, 59397, 64080, 69003, 74172

Offset: 0

N. J. A. Sloane, Apr 16 2000

Every term is the product plus the sum of 3 consecutive numbers. - Vladimir Joseph Stephan Orlovsky, Oct 24 2009

Continued fraction [n,n,n] = (n^2+1)/(n^3+2n) = (n^2+1)/a(n); e.g., [7,7,7] = 50/357. - Gary W. Adamson, Jul 15 2010

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Thomas Oléron Evans, Queues of Cubes, Mathistopheles, August 22 2015.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), pp. 37-42.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A185651.

Cf. A006527, A007531, A008585, A073133, A292022.

nterms=100;Table[n^3+2n,{n,0,nterms}] (* Paolo Xausa, Nov 25 2021 *)

a(n)=n^3+2*n \\ Charles R Greathouse IV, Sep 01 2015

a(n) = n^3 + 2*n = A073133(n,3). - Henry Bottomley, Jul 16 2002
G.f.: 3*x*(x^2+1)/(x-1)^4. - Colin Barker, Dec 21 2012
a(n) = ((n-1)^3 + n^3 + (n+1)^3)/3. - David Morales Marciel, Aug 28 2015
From Bernard Schott, Nov 28 2021: (Start)
a(n) = A007531(n+1) + A008585(n) (see 1st comment).
a(n) = 3*A006527(n). (End)
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: exp(x)*x*(3 + 3*x + x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A292022(n)/4. (End)

A128960 a(n) = (n^3 - n)*2^n.

Original entry on oeis.org

0, 24, 192, 960, 3840, 13440, 43008, 129024, 368640, 1013760, 2703360, 7028736, 17891328, 44728320, 110100480, 267386880, 641728512, 1524105216, 3586129920, 8367636480, 19377684480, 44568674304, 101871255552, 231525580800, 523449139200, 1177760563200, 2638183661568

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A000079, A007531, A036289, A128796.

Cf. A128961, A128962, A128963, A128964, A128965, A128967, A128969.

[(n^3-n)*2^n: n in [1..25]]; /* or */ I:=[0,24,192,960]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[24 x/(1 - 2 x)^4, {x, 0, 30}], x] (* or *) LinearRecurrence[{8, -24, 32, -16}, {0, 24, 192, 960}, 30] (* Vincenzo Librandi, Feb 12 2013 *)

a(n)=(n^3-n)<Charles R Greathouse IV, Oct 07 2015

G.f.: 24*x^2/(1-2*x)^4. - Vincenzo Librandi, Feb 12 2013
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4). - Vincenzo Librandi, Feb 12 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A000079(n).
Sum_{n>=2} 1/a(n) = (2*log(2)-1)/8.
Sum_{n>=2} (-1)^n/a(n) = (3/2)^2*log(3/2) - 7/8. (End)

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset: 0

Peter Luschny, Dec 19 2015

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
From Antti Karttunen, Dec 19 2015: (Start)
Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

			Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
Index entries for sequences related to factorial numbers

Triangle giving terms only up to column k=n: A124320.
Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
Row sums of the triangle: A000522.
A(n, n) = A000407(n-1) for n>0.
2^n*A(1/2,n) = A001147(n).
Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])

(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015
E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018
A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022
G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

A368437 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = 2n+1-k, where (x,y,z) is a permutation of three distinct numbers taken from {0,1,...,n}, for n >= 2, k >= 2.

Original entry on oeis.org

4, 2, 4, 4, 12, 4, 4, 4, 12, 14, 20, 6, 4, 4, 12, 12, 28, 24, 28, 8, 4, 4, 12, 12, 24, 30, 44, 34, 36, 10, 4, 4, 12, 12, 24, 24, 48, 48, 60, 44, 44, 12, 4, 4, 12, 12, 24, 24, 40, 50, 72, 66, 76, 54, 52, 14, 4, 4, 12, 12, 24, 24, 40, 40, 72, 76, 96, 84, 92

Offset: 1

Clark Kimberling, Dec 25 2023

Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!. The limiting row, (4, 4, 12, 12, 24, 24, 40, 40, ...) consists of repeated terms of (A046092(n+1)) = (4, 12, 24, 40, ...).

			Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = 2n+1-k, for k = 2,3, are as follows:
012: |0-1| + |1-2| = 2
021: |0-2| + |2-1| = 3
102: |1-0| + |0-2| = 3
120: |1-2| + |2-0| = 3
201: |2-0| + |0-1| = 3
210: |2-1| + |1-0| = 2
so that row 1 of the array is (4,2), representing four 2s and two 3s.
First eight rows:
  4  2
  4  4  12   4
  4  4  12  14  20   6
  4  4  12  12  28  24  28   8
  4  4  12  12  24  30  44  34  36  10
  4  4  12  12  24  24  48  48  60  44  44  12
  4  4  12  12  24  24  40  50  72  66  76  54  52  14
  4  4  12  12  24  24  40  40  72  76  96  84  92  64  60  16

Cf. A007531, A046092, A368435, A368436.

t[n_] := t[n] = Permutations[-1 + Range[n + 1], {3}];
a[n_, k_] :=  Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == 2n+1-k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 1}];
v = Flatten[u] (* sequence *)
Column[u]      (* array *)

A279019 Least possible number of diagonals of simple convex polyhedron with n faces.

Original entry on oeis.org

0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset: 4

Vladimir Letsko, Dec 03 2016

Obviously, a pyramid has no diagonals. Hence minimum of diagonals of an arbitrary convex polyhedron having n faces is equal to 0.
Minimum number of diagonals among simple convex polyhedra having n faces is obtained from a polyhedron with two triangular faces, n-4 quadrangular faces and two (n-1)-sided faces. A polyhedron having 3 triangular faces, 3 pentagonal faces and 1 hexagonal face gives another example of a simple convex polyhedron with the least possible number of diagonals for n = 7. A polyhedron having 4 triangular faces and 4 hexagonal faces gives a similar example for n = 8.
Essentially the same as A103505 and A002378. - R. J. Mathar, Dec 05 2016

Colin Barker, Table of n, a(n) for n = 4..1000
David Bremner and Victor Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000944, A002378, A279015, A279022, A007531.

Table[(n-4)(n-5),{n,4,60}] (* or *) LinearRecurrence[{3,-3,1},{0,0,2},60] (* Harvey P. Dale, Sep 23 2019 *)

concat(vector(2), Vec(2*x^6 / (1 - x)^3 + O(x^60))) \\ Colin Barker, Dec 05 2016

a(n) = n^2 - 9*n + 20 = (n-4)*(n-5).
G.f.: -2*x^6/(x-1)^3. - R. J. Mathar, Dec 05 2016
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6. - Colin Barker, Dec 05 2016
E.g.f.: exp(x)*(20 - 8*x + x^2) - x^3/3 - 3*x^2 - 12*x - 20. - Stefano Spezia, Nov 24 2019
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=6} 1/a(n) = 1.
Sum_{n>=6} (-1)^n/a(n) = 2*log(2) - 1. (End)

A268944 T(n,k)=Number of length-n 0..k arrays with no repeated value unequal to the previous repeated value plus one mod k+1.

Original entry on oeis.org

2, 3, 4, 4, 9, 6, 5, 16, 24, 10, 6, 25, 60, 63, 14, 7, 36, 120, 220, 159, 22, 8, 49, 210, 565, 788, 396, 30, 9, 64, 336, 1206, 2615, 2780, 969, 46, 10, 81, 504, 2275, 6834, 11950, 9684, 2349, 62, 11, 100, 720, 3928, 15239, 38322, 54045, 33404, 5640, 94, 12, 121, 990

Offset: 1

R. H. Hardin, Feb 16 2016

Table starts
..2.....3......4.......5........6.........7.........8..........9.........10
..4.....9.....16......25.......36........49........64.........81........100
..6....24.....60.....120......210.......336.......504........720........990
.10....63....220.....565.....1206......2275......3928.......6345.......9730
.14...159....788....2615.....6834.....15239.....30344......55503......95030
.22...396...2780...11950....38322....101192....232696.....482490.....923150
.30...969...9684...54045...213042....667065...1773384....4171869....8925990
.46..2349..33404..242365..1175850...4370261..13443064...35904789...85953830
.62..5640.114292.1079240..6450402..28480312.101433800..307754712..824720230
.94.13455.388444.4777225.35200458.184750699.762265720.2628421029.7887767350

			Some solutions for n=6 k=4
..2. .4. .3. .1. .4. .4. .1. .2. .2. .0. .4. .2. .0. .3. .2. .3
..1. .4. .3. .0. .1. .0. .2. .3. .4. .0. .0. .1. .3. .0. .2. .3
..1. .2. .2. .1. .3. .4. .4. .0. .2. .2. .2. .2. .0. .2. .3. .1
..2. .4. .4. .3. .4. .0. .1. .2. .4. .1. .1. .3. .1. .0. .4. .2
..3. .0. .2. .0. .2. .1. .2. .3. .1. .0. .0. .0. .1. .3. .0. .0
..2. .3. .0. .0. .4. .4. .0. .4. .2. .2. .1. .2. .4. .3. .2. .4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A027383.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A007531(n+2).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3)
k=3: a(n) = 5*a(n-1) -2*a(n-2) -12*a(n-3)
k=4: a(n) = 7*a(n-1) -7*a(n-2) -20*a(n-3)
k=5: a(n) = 9*a(n-1) -14*a(n-2) -30*a(n-3)
k=6: a(n) = 11*a(n-1) -23*a(n-2) -42*a(n-3)
k=7: a(n) = 13*a(n-1) -34*a(n-2) -56*a(n-3)
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 2*n
n=4: a(n) = n^4 + 4*n^3 + 3*n^2 + n + 1
n=5: a(n) = n^5 + 5*n^4 + 4*n^3 + 3*n^2 + 2*n - 1
n=6: a(n) = n^6 + 6*n^5 + 5*n^4 + 6*n^3 + 3*n^2 - n + 2
n=7: a(n) = n^7 + 7*n^6 + 6*n^5 + 10*n^4 + 4*n^3 + n^2 + 4*n - 3

cp's OEIS Frontend

A182797 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.

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A045619 Numbers that are the products of 2 or more consecutive integers.

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A052787 Product of 5 consecutive integers.

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A055112 a(n) = n*(n+1)*(2*n+1).

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A054602 a(n) = Sum_{d|3} phi(d)*n^(3/d).

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A128960 a(n) = (n^3 - n)*2^n.

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A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

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A055112 a(n) = n(n+1)(2*n+1).