A083374 -id:A083374 - cp's OEIS frontend

A240444 Triangle T(n, k) = Number of ways to arrange k indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

1, 1, 1, 4, 6, 4, 1, 9, 36, 84, 120, 96, 32, 1, 16, 120, 560, 1800, 4128, 6726, 7492, 5238, 1924, 232, 1, 25, 300, 2300, 12600, 52080, 166702, 416622, 808488, 1196196, 1306464, 1001364, 497940, 141336, 18208, 636, 1, 36, 630, 7140, 58800, 373632, 1895938, 7835492

Offset: 1

Heinrich Ludwig, May 07 2014

The triangle is irregularly shaped: 0 <= k <= A240443(n). The first row corresponds to n = 1.

The maximal number of points that can be placed on an n X n square grid so that no four points are vertices of a square is A240443(n).

			The triangle begins:
  1,  1;
  1,  4,   6,   4;
  1,  9,  36,  84,  120,   96,   32;
  1, 16, 120, 560, 1800, 4128, 6726, 7492, 5238, 1924, 232;
...

Heinrich Ludwig, Table of n, a(n) for n = 1..101

Cf. A240443, A000290 (column 2), A083374 (column 3), A178208 (column 4), A006857 (column 5 divided by 120), A240445 (column 6), A240446 (column 7).

A253650 Triangular numbers that are the product of a triangular number and a square number (both greater than 1).

Original entry on oeis.org

300, 1176, 3240, 7260, 14196, 25200, 29403, 41616, 64980, 97020, 139656, 195000, 228150, 265356, 353220, 461280, 592416, 749700, 936396, 1043290, 1155960, 1412040, 1708476, 2049300, 2438736, 2881200, 3381300, 3499335, 3943836, 4573800, 5276376, 6056940, 6921060, 7874496

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

			3240 is in the sequence because 3240 is triangular number (3240=80*81/2), and 3240=10*324=(4*5/2)*(18^2), product of triangular number 10 and square number 324.

Charles R Greathouse IV, Table of n, a(n) for n = 1..3486

Cf. A188630, A083374, A185096, A253651, A253652, A253653.

triQ[n_] := IntegerQ@ Sqrt[8n + 1]; lst = Sort@ Flatten@ Outer[Times, Table[ n(n + 1)/2, {n, 2, 400}], Table[ n^2, {n, 2, 200}]]; Select[ lst, triQ] (* Robert G. Wilson v, Jan 13 2015 *)

{i=3; j=3; while(i<=10^7, k=3; p=3; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}

is(n)=if(!ispolygonal(n,3), return(0)); fordiv(core(n,1)[2], d, d>1 && ispolygonal(n/d^2,3) && n>d^2 && return(1)); 0 \\ Charles R Greathouse IV, Sep 29 2015

list(lim)=my(v=List(),t,c); for(n=24,(sqrt(8*lim+1)-1)\2, t=n*(n+1)/2; c=core(n,1)[2]*core(n+1,1)[2]; if(valuation(t,2)\2 < valuation(c,2), c/=2); fordiv(c, d, if(d>1 && ispolygonal(t/d^2,3) && t>d^2, listput(v,t); break))); Vec(v) \\ Charles R Greathouse IV, Sep 29 2015

A241219 Number of ways to choose two points on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 21, 171, 666, 1830, 4095, 8001, 14196, 23436, 36585, 54615, 78606, 109746, 149331, 198765, 259560, 333336, 421821, 526851, 650370, 794430, 961191, 1152921, 1371996, 1620900, 1902225, 2218671, 2573046, 2968266, 3407355, 3893445, 4429776, 5019696, 5666661

Offset: 1

Martin Renner, Apr 17 2014

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

a(n) is also the number of segments on a centered hexagonal grid of size n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hex Number.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A083374.

[Binomial(3*n^2-3*n+1, 2): n in [1..35]]; // Vincenzo Librandi, Apr 19 2014

seq(binomial(3*n^2-3*n+1, 2),n=1..34); # Martin Renner, Apr 27 2014
op(PolynomialTools[CoefficientList](convert(series(-3*x^2*(7*x^2+22*x+7)/(x-1)^5, x=0, 35), polynom), x)[2..35]); # Martin Renner, Apr 27 2014

CoefficientList[Series[-3 x^2 (7 x^2 + 22 x + 7)/(x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 19 2014 *)

concat(0, Vec(-3*x^2*(7*x^2+22*x+7) / (x-1)^5 + O(x^100))) \\ Colin Barker, Apr 18 2014

a(n) = binomial(A003215(n-1), 2).
     = binomial(3*n^2-3*n+1, 2).
     = 3/2*n*(n-1)*(3*n^2-3*n+1).
     = 9/2*n^4-9*n^3+6*n^2-3/2*n.
G.f.: -3*x^2*(7*x^2+22*x+7) / (x-1)^5. - Colin Barker, Apr 18 2014
Sum_{n>=2} 1/a(n) = 8/3 - 2*Pi*tanh(Pi/(2*sqrt(3)))/sqrt(3). - Amiram Eldar, Feb 17 2024

Typo in Mathematica program fixed by Martin Renner, Apr 27 2014

A243645 Number of ways two L-tiles can be placed on an n X n square.

Original entry on oeis.org

0, 0, 0, 1, 20, 87, 244, 545, 1056, 1855, 3032, 4689, 6940, 9911, 13740, 18577, 24584, 31935, 40816, 51425, 63972, 78679, 95780, 115521, 138160, 163967, 193224, 226225, 263276, 304695, 350812, 401969, 458520, 520831, 589280, 664257, 746164, 835415, 932436

Offset: 0

Alois P. Heinz, Jun 08 2014

This sequence also represents the number of edges added to G so that it is complete, where G is a graph of (n-1)^2 nodes arranged in a rhombus and embedded in the hexagonal lattice. G begins with A045944(n-2) edges and a(n) edges are added to form a complete graph. - John Tyler Rascoe, Sep 24 2022

			a(3) = 1:
._____.
|_| |_|
| |___|
|___|_| .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column k=2 of A243608.

a:= n-> `if`(n<2, 0, ((((n-4)*n-1)*n+18)*n-16)/2):
seq(a(n), n=0..50);

CoefficientList[Series[x^3 (x^3+3x^2-15x-1)/(x-1)^5,{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,1,20,87,244},40] (* Harvey P. Dale, Mar 06 2016 *)

G.f.: x^3*(x^3+3*x^2-15*x-1) / (x-1)^5.
a(n) = (n^4-4*n^3-n^2+18*n-16)/2 for n>=2, a(n) = 0 for n<2.
a(n) = A083374(n-1) - A045944(n-2) for n>=2. - John Tyler Rascoe, Sep 24 2022

A253285 a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.

Original entry on oeis.org

0, 24, 240, 1200, 4200, 11760, 28224, 60480, 118800, 217800, 377520, 624624, 993720, 1528800, 2284800, 3329280, 4744224, 6627960, 9097200, 12289200, 16364040, 21507024, 27931200, 35880000, 45630000, 57493800, 71823024, 89011440, 109498200, 133771200, 162370560

Offset: 0

Peter Luschny, Mar 23 2015

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A006542, A083374, A133754, A253284.

List([0..40], n -> n*((n+1)*(n+2))^2*(n+3)/6); # Bruno Berselli, Mar 06 2018

[n*((n+1)*(n+2))^2*(n+3)/6: n in [0..40]]; // Bruno Berselli, Mar 06 2018

Maple
```
seq(n*((n+1)*(n+2))^2*(n+3)/6,n=0..19);
```

Table[n ((n + 1) (n + 2))^2 (n + 3)/6, {n, 0, 40}] (* Bruno Berselli, Mar 06 2018 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,24,240,1200,4200,11760,28224},40] (* Harvey P. Dale, Aug 05 2024 *)

[n*((n+1)*(n+2))**2*(n+3)/6 for n in range(40)] # Bruno Berselli, Mar 06 2018

[n*((n+1)*(n+2))^2*(n+3)/6 for n in (0..40)] # Bruno Berselli, Mar 06 2018

G.f.: -24/(x-1)^4 - 144/(x-1)^5 - 240/(x-1)^6 - 120/(x-1)^7. See the comment in A253284 for the general case.
a(n) = n*((n+1)*(n+2))^2*(n+3)/6.
a(n) = (N^3 + 4*N^2 + 4*N)/6 = N*(N + 2)^2/6 with N = n^2 + 3*n.
From Bruno Berselli, Mar 06 2018: (Start)
a(n) = 24*A006542(n+3) for n>0.
a(n) = Sum_{i=0..n} i*(i+1)^3*(i+2). Therefore, the first differences are in A133754. (End)

A253652 Triangular numbers that are the product of a triangular number and an oblong number.

Original entry on oeis.org

0, 6, 36, 120, 210, 300, 630, 1176, 2016, 3240, 3570, 4950, 7140, 7260, 10296, 14196, 19110, 23436, 25200, 32640, 39060, 41616, 52326, 61776, 64980, 79800, 97020, 116886, 139656, 145530, 165600, 195000, 228150, 242556, 265356, 304590, 306936, 349866, 353220, 404550, 426426, 461280

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

Supersequence of A083374, because A083374(n)= n^2 * (n^2 - 1)/2 = n*(n+1)/2*n*(n-1), product of triangular number n*(n+1)/2 and oblong number n*(n-1).

			630 is in the sequence because it is a triangular number (630 = 35*36/2) and 630 = 105*6, with 105 = 14*15/2, triangular number, and 6 = 2*3, oblong number.

Cf. A002378, A000217, A188630, A083374, A253650, A253651, A253653.

{i=0;j=1;print1(0,", ");while(i<=10^6,k=1;p=2;c=0;while(k0,c=k);if(c>0,print1(i,", "));k+=p;p+=1);i+=j;j+=1)}

A254881 Triangle read by rows, T(n,k) = sum(j=0..k-1, S(n+1,j+1)*S(n,k-j)) where S denotes the Stirling cycle numbers A132393, T(0,0)=1, n>=0, 0<=k<=2n.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 12, 40, 51, 31, 9, 1, 0, 144, 564, 904, 769, 376, 106, 16, 1, 0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1, 0, 86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36, 1, 0, 3628800, 18299520

Offset: 0

Peter Luschny, Feb 10 2015

These are also the coefficients of the polynomials interpolating the sequence k -> n!*((n+k)!/k!)*binomial(n+k-1,k-1) (for fixed n>=0). Divided by n! these polynomials generate the rows of Lah numbers L(n+k, k) = ((n+k)!/k!)* binomial(n+k-1,k-1).

			[1]
[0, 1, 1]
[0, 2, 5, 4, 1]
[0, 12, 40, 51, 31, 9, 1]
[0, 144, 564, 904, 769, 376, 106, 16, 1]
[0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1]
For example in the case n=3 the polynomial (k^6+9*k^5+31*k^4+51*k^3+40*k^2+12*k)/3! generates the Lah numbers 0, 24, 240, 1200, 4200, 11760, 28224, ... (A253285).

Cf. A246117, A254882, A187235.

The sequences A000012, A002378, A083374, A253285 are the Lah number rows generated by the polynomials divided by n! for n=0, 1, 2, 3 respectivly.

# This is a special case of the recurrence given in A246117.
t := proc(n,k) option remember; if n=0 and k=0 then 1 elif
k <= 0 or k>n then 0 else iquo(n,2)*t(n-1,k)+t(n-1,k-1) fi end:
A254881 := (n,k) -> t(2*n,k):
seq(print(seq(A254881(n,k), k=0..2*n)), n=0..5);
# Illustrating the comment:
restart: with(PolynomialTools): with(CurveFitting): for N from 0 to 5 do
CoefficientList(PolynomialInterpolation([seq([k,N!*((N+k)!/k!)*binomial(N+k-1,k-1)], k=0..2*N)], n), n) od;

Flatten[{1,Table[Table[Sum[Abs[StirlingS1[n+1,j+1]] * Abs[StirlingS1[n,k-j]],{j,0,k-1}],{k,0,2*n}],{n,1,10}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

def T(n,k):
    if n == 0: return 1
    return sum(stirling_number1(n+1,j+1)*stirling_number1(n,k-j) for j in range(k))
for n in range (6): [T(n,k) for k in (0..2*n)]

T(n, n) = A187235(n) for n>=1 (after the explicit formula of Vaclav Kotesovec).

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

Original entry on oeis.org

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

Offset: 3

Stefano Spezia, Mar 19 2021

			The array begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  21   45   78  120  171 ...
4  |  36   78  136  210  300 ...
5  |  55  120  210  325  465 ...
6  |  78  171  300  465  666 ...
7  | 105  231  406  630  903 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten

O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

A368831 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the n X n rook graph (n >= 0, 0 <= k <= n^2).

Original entry on oeis.org

1, 0, 1, 0, 0, 6, 4, 1, 0, 0, 0, 48, 117, 126, 84, 36, 9, 1, 0, 0, 0, 0, 488, 2640, 6712, 10864, 12726, 11424, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 6130, 58300, 269500, 808325, 1778875, 3075160, 4349400, 5154900, 5186300, 4454400, 3268360, 2042950, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset: 0

Stephan Mertens, Jan 07 2024

The entries in row n are the coefficients of the domination polynomial of the n X n rook graph.
Sum of entries in row n = A287065 = main diagonal of A287274.
Number of minimum dominating sets T(n,n) = A248744(n).

			Triangle begins: (first 5 rows)
  1;
  0, 1;
  0, 0, 6,  4,   1;
  0, 0, 0, 48, 117,  126,   84,    36,     9,     1;
  0, 0, 0,  0, 488, 2640, 6712, 10864, 12726, 11424, 8008, 4368, 1820, 560, 120, 16, 1;
  ...

John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, chapter 7.

Alois P. Heinz, Rows n = 0..32, flattened
Stephan Mertens, Domination Polynomial of the Rook Graph, Journal of Integer Sequences 27 (2024), Article 24.3.7; arXiv:2401.00716 [math.CO], 2024.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Cf. A000290, A083374, A287065 (row sums), A287274, A248744 (leading diagonal).

R[n_, m_] := CoefficientList[((x + 1)^n - 1)^m - (-1)^m*Sum[Binomial[m, k]*(-1)^k*((1 + x)^k - 1)^n, {k, 0, m - 1}], x];
Flatten[Table[R[n,n],{n,1,5}]]

G.f.: ((x+1)^n - 1)^m - (-1)^m * Sum_{k=0..m-1} binomial(m,k)*(-1)^k*((1+x)^k - 1)^n (for the rectangular n X m rook graph).

T(n,n) = 2*n^n - n!.

A227970 Triangular arithmetic on half-squares: b(n)*(b(n) - 1)/2 where b(n) = floor(n^2/2).

Original entry on oeis.org

0, 0, 1, 6, 28, 66, 153, 276, 496, 780, 1225, 1770, 2556, 3486, 4753, 6216, 8128, 10296, 13041, 16110, 19900, 24090, 29161, 34716, 41328, 48516, 56953, 66066, 76636, 87990, 101025, 114960, 130816, 147696, 166753, 186966, 209628, 233586, 260281, 288420, 319600, 352380, 388521, 426426

Offset: 0

Richard R. Forberg, Aug 01 2013

Analogous to A083374 for the squares A000290.

A "mirrored" repeating pattern of cycle length 20 exists in the last digit.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A007590, A083374.

A227970:=n->binomial(floor(n^2/2),2); seq(A227970(k),k=0..50); # Wesley Ivan Hurt, Oct 05 2013

Table[Binomial[Floor[n^2/2], 2], {n, 0, 50}] (* Wesley Ivan Hurt, Sep 27 2013 *)

Let b(n) = floor(n^2/2), for n => 0, then a(n) = b(n)*(b(n)- 1)/2.
G.f.: -x^2*(1+4*x+14*x^2+4*x^3+x^4) / ( (1+x)^3*(x-1)^5 ). - R. J. Mathar, Aug 14 2013
a(n) = binomial(floor(n^2/2), 2). - Wesley Ivan Hurt, Sep 27 2013

cp's OEIS Frontend

A240444 Triangle T(n, k) = Number of ways to arrange k indistinguishable points on an n X n square grid so that no four of them are vertices of a square of any orientation.

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A253650 Triangular numbers that are the product of a triangular number and a square number (both greater than 1).

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Mathematica

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A241219 Number of ways to choose two points on a centered hexagonal grid of size n.

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A243645 Number of ways two L-tiles can be placed on an n X n square.

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A253285 a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.

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A253652 Triangular numbers that are the product of a triangular number and an oblong number.

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PARI

A254881 Triangle read by rows, T(n,k) = sum(j=0..k-1, S(n+1,j+1)*S(n,k-j)) where S denotes the Stirling cycle numbers A132393, T(0,0)=1, n>=0, 0<=k<=2n.

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