A037270 -id:A037270 - cp's OEIS frontend

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.

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.

A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.

Original entry on oeis.org

1, 34, 189, 616, 1525, 3186, 5929, 10144, 16281, 24850, 36421, 51624, 71149, 95746, 126225, 163456, 208369, 261954, 325261, 399400, 485541, 584914, 698809, 828576, 975625, 1141426, 1327509, 1535464, 1766941, 2023650, 2307361, 2619904, 2963169, 3339106, 3749725, 4197096

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Partial sums of centered dodecahedral numbers (A005904).

OEIS Wiki, Centered Platonic numbers
Eric Weisstein's World of Mathematics, Platonic Solid
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A005904, A006566, A008354, A014820, A037270, A132366.

[(n + 1)^2*(5*n^2 + 10*n + 2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 15 2017

A269237:=n->(n + 1)^2*(5*n^2 + 10*n + 2)/2: seq(A269237(n), n=0..50); # Wesley Ivan Hurt, Oct 15 2017

Table[(n + 1)^2 ((5 n^2 + 10 n + 2)/2), {n, 0, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 34, 189, 616, 1525}, 36]

x='x+O('x^99); Vec((1+29*x+29*x^2+x^3)/(1-x)^5) \\ Altug Alkan, Apr 10 2016

G.f.: (1 + 29*x + 29*x^2 + x^3)/(1 - x)^5.
E.g.f.: exp(x)*(2 + 66*x + 122*x^2 + 50*x^3 + 5*x^4)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
Sum_{n>=0} 1/a(n) = (5 - Pi^2 - sqrt(15)*Pi*cot(sqrt(3/5)*Pi))/9 = 1.0377796966... . - Vaclav Kotesovec, Apr 10 2016

A321230 Number of set partitions of [n^2] into n n-element subsets having the same sum.

Original entry on oeis.org

1, 1, 1, 2, 392, 3245664, 6534071578530

Offset: 0

Alois P. Heinz, Oct 31 2018

			a(0) = 1: empty.
a(1) = 1: 1.
a(2) = 1: 14|23.
a(3) = 2: 168|249|357, 159|267|348.

Wikipedia, Partition of a set

Main diagonal of A203986.

Cf. A006003, A037270, A052456, A104185, A321282.

a(n) = A203986(n,n).

A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

Original entry on oeis.org

0, 1, 21, 140, 575, 1785, 4606, 10416, 21330, 40425, 71995, 121836, 197561, 308945, 468300, 690880, 995316, 1404081, 1943985, 2646700, 3549315, 4694921, 6133226, 7921200, 10123750, 12814425, 16076151, 20001996

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each member of a chiral pair is a reflection, but not a rotation, of the other. A regular tetrahedron has 6 edges and Schläfli symbol {3,3}.

			For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.

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

Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).
Other elements: A000332 (vertices and faces).
Other polyhedra: A337406 (cube/octahedron).
Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).

Mathematica
```
Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]
```

a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.
a(n) = 1*C(n,2) + 18*C(n,3) + 62*C(n,4) + 75*C(n,5) + 30*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).
G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.

A373329 a(n)^2 is the greatest square not exceeding A000217(n^2).

Original entry on oeis.org

0, 1, 3, 6, 11, 18, 25, 35, 45, 57, 71, 85, 102, 119, 138, 159, 181, 204, 229, 255, 283, 312, 342, 374, 407, 442, 478, 515, 554, 595, 636, 679, 724, 770, 817, 866, 916, 968, 1021, 1075, 1131, 1189, 1247, 1307, 1369, 1432, 1496, 1562, 1629, 1698, 1768, 1839, 1912

Offset: 0

Hugo Pfoertner, Jun 01 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A001109 (subsequence), A001110, A001333, A037270, A061288.

a:= n-> floor(sqrt((t-> t*(t+1)/2)(n^2))):
seq(a(n), n=0..52);  # Alois P. Heinz, Jun 01 2024

Array[Floor@ Sqrt[(#^4 + #^2)/2] &, 53, 0] (* Michael De Vlieger, Jun 02 2024 *)

PARI
```
a(n) = sqrtint((n^4+n^2)/2)
```

a(n) = A061288(n^2). - Alois P. Heinz, Jun 01 2024

A363460 a(n) is the permanent of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.

Original entry on oeis.org

1, 1, 11, 556, 74964, 21700112, 11500685084, 10057140949968, 13496937368200000, 26331147893897760544, 71606290155732170272320, 262516365211410942628577408, 1262517559940020030446967822592, 7786463232979127181938238723356160, 60414239829783205320232261233394491136

Offset: 0

Stefano Spezia, Jun 03 2023

nonn

The matrix is the upper-left n X n part of the square arrangement in A081344.

The matrix element k is at row A220604(k) and column A220603(k), for k = 1..n^2.

			a(5) = 21700112 is the permanent of the 5 X 5 matrix
  |  1----2    9---10   25 |
  |       |    |    |    | |
  |  4----3    8   11   24 |
  |  |         |    |    | |
  |  5----6----7   12   23 |
  |                 |    | |
  | 16---15---14---13   22 |
  |  |                   | |
  | 17---18---19---20---21 |

Cf. A006527 (trace), A037270 (elements sum of the matrix), A060736, A061349 (anti trace), A081344, A220603, A220604, A363376 (determinant).

a={1}; For[n=1, n<=14, n++,k=i=j=1; M[i,j]=k++; For[h=1, h

A051672 Triangle of up-down sums of k-th powers: a(n,k)=sum(i^k,i=1..n)+sum((n-i)^k,i=1..n-1), n,k>0.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 19, 10, 1, 25, 44, 45, 18, 1, 36, 85, 136, 115, 34, 1, 49, 146, 325, 452, 309, 66, 1, 64, 231, 666, 1333, 1576, 859, 130, 1, 81, 344, 1225, 3254, 5725, 5684, 2445, 258, 1, 100, 489, 2080, 6951, 16626, 25405, 21016, 7075, 514, 1, 121

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

			{1}; {4,1}; {9,6,1}; {16,19,10,1}; {25,44,45,18,1}; ...

Cf. A000290, A005900, A037270.

a[n_, k_] := HarmonicNumber[n, -k]+Zeta[-k]-Zeta[-k, n]; Flatten[ Table[ a[n-k+1, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Nov 29 2011 *)

a(n, 1)=n^2=A000290, a(n, 2)=1/3*n*(2*n^2+1)=A005900, a(n, 3)= (1/2) *n^2*(n^2+1)=A037270, a(n, 4)=1/15*n*(6*n^4+10*n^2-1), a(n, 5)=1/6*n^2*(2*n^4+5*n^2-1)

A094260 Sum of next n numbers/n if n divides the sum else n times the sum of next n numbers.

Original entry on oeis.org

1, 10, 5, 136, 13, 666, 25, 2080, 41, 5050, 61, 10440, 85, 19306, 113, 32896, 145, 52650, 181, 80200, 221, 117370, 265, 166176, 313, 228826, 365, 307720, 421, 405450, 481, 524800, 545, 668746, 613, 840456, 685, 1043290, 761, 1280800, 841, 1556730, 925, 1875016, 1013, 2239786

Offset: 1

Amarnath Murthy, Apr 26 2004

Quasipolynomial of order 2 and degree 5. - Charles R Greathouse IV, Oct 14 2013

			The sequence is: 1/1, (2+3)*2, (4+5+6)/3, (7+8+9+10)*4, ...

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000217, A037270, A001844.

LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{1,10,5,136,13,666,25,2080,41,5050},50] (* Harvey P. Dale, May 01 2020 *)
fix[c_]:=If[Mod[Total[c],Length[c]]==0,Total[c]/Length[c],Length[c] Total[c]]; fix/@With[ {nn=50},TakeList[ Range[(nn(nn+1))/2],Range[nn]]] (* Harvey P. Dale, Apr 05 2023 *)

a(n) = if (n%2, (n^2+1)/2, n^2*(n^2+1)/2); \\ Michel Marcus, Aug 23 2022

For even n, a(n) = A000217(n^2) = n^2*(n^2+1)/2; for odd n, a(n) = (n^2 + 1)/2.

Sum_{n>=1} 1/a(n) = 1 + Pi^2/12 - Pi*cosech(Pi). - Amiram Eldar, Aug 23 2022

Edited and extended by Max Alekseyev, Apr 26 2009

A342797 Irregular triangle read by rows: T(n, k) is the k-th antidiagonal sum of the n X n matrices defined in A069480 and A078475.

Original entry on oeis.org

1, 1, 5, 4, 1, 5, 15, 15, 9, 1, 5, 15, 34, 36, 29, 16, 1, 5, 15, 34, 65, 70, 63, 47, 25, 1, 5, 15, 34, 65, 111, 120, 114, 96, 69, 36, 1, 5, 15, 34, 65, 111, 175, 189, 185, 166, 135, 95, 49, 1, 5, 15, 34, 65, 111, 175, 260, 280, 279, 260, 226, 180, 125, 64

Offset: 1

Stefano Spezia, Apr 25 2021

			The triangle T(n, k) begins:
1
1    5    4
1    5   15   15    9
1    5   15   34   36   29   16
1    5   15   34   65   70   63   47   25
...

Cf. A000290 (diagonal), A006003, A037270 (row sums), A060747 (row length), A069480, A078475.

T[n_,k_]:=If[k<=n,(k+k^3)/2,(k^3+2n-6k^2n-4n^3+k(10n^2-1))/2]; Flatten[Table[T[n,k],{n,8},{k,2n-1}]]

T(n, k) = A006003(k) for 1 <= k <= n.
T(n, k) = (k^3 + 2*n - 6*k^2*n - 4*n^3 + k*(10*n^2 - 1))/2 for n < k <= 2*n - 1.
T(n, 2*n-1) = A000290(n).

A343155 Irregular triangle T read by rows: T(n, k) is the sum of the consecutive integers placed along the k-th turn of the spiral of the n X n matrix defined in A126224.

Original entry on oeis.org

1, 10, 36, 9, 78, 58, 136, 164, 25, 210, 318, 138, 300, 520, 356, 49, 406, 770, 654, 250, 528, 1068, 1032, 612, 81, 666, 1414, 1490, 1086, 394, 820, 1808, 2028, 1672, 932, 121, 990, 2250, 2646, 2370, 1614, 570, 1176, 2740, 3344, 3180, 2440, 1316, 169, 1378, 3278, 4122, 4102, 3410, 2238, 778

Offset: 1

Stefano Spezia, Apr 07 2021

			The triangle T(n, k) begins:
n\k|   1    2    3    4
---+-------------------
1  |   1
2  |  10
3  |  36    9
4  |  78   58
5  | 136  164   25
6  | 210  318  138
7  | 300  520  356   49
...
For n = 1 the matrix is
      1
and T(1, 1) = 1.
For n = 2 the matrix is
      1, 2
      4, 3
and T(2, 1) = 1 + 2 + 3 + 4 = 4*5/2 = 10.
For n = 3 the matrix is
      1, 2, 3
      8, 9, 4
      7, 6, 5
and T(3, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 8*9/2 = 36; T(3, 2) = 9.
For n = 4 the matrix is
      1,  2,  3,  4
     12, 13, 14,  5
     11, 16, 15,  6
     10,  9,  8,  7
and T(4, 1) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 12*13/2 = 78; T(4, 2) = 13 + 14 + 15 + 16 = (13 + 16)*4/2 = 58.
...

Cf. A000007, A000290, A033585, A037270 (row sums), A110654, A126224.

Table[2(2k-n-1)(3+8k(k-n-1)+4n)+n^2KroneckerDelta[n,2k-1],{n,14},{k,Ceiling[n/2]}]//Flatten

T(n, k) = 2*(2*k - n - 1)*(3 + 8*k*(k - n - 1) + 4*n) + n^2*0^(n+1-2*k) with 0 < k <= ceiling(n/2).

T(n, 1) = A033585(n-1) for n > 1.

cp's OEIS Frontend

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.

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A269237 a(n) = (n + 1)^2*(5*n^2 + 10*n + 2)/2.

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A321230 Number of set partitions of [n^2] into n n-element subsets having the same sum.

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A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

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Mathematica

Formula

A373329 a(n)^2 is the greatest square not exceeding A000217(n^2).

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Maple

Mathematica

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A363460 a(n) is the permanent of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.

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Mathematica

A051672 Triangle of up-down sums of k-th powers: a(n,k)=sum(i^k,i=1..n)+sum((n-i)^k,i=1..n-1), n,k>0.

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A094260 Sum of next n numbers/n if n divides the sum else n times the sum of next n numbers.

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A269237 a(n) = (n + 1)^2(5n^2 + 10*n + 2)/2.