A120070 -id:A120070 - cp's OEIS frontend

A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

1, 1, 7, 100, 2840, 129428, 8613997, 791557152, 95921167710, 14818153059968, 2842735387366627, 663020104070865664, 184757202542187563476, 60623405966739216871680, 23135486197103263598936745, 10160292704659539620791062528, 5087671168376607498331875818106

Offset: 0

Stefano Spezia, Dec 30 2018

nonn

The matrix M(n) differs from that of A306457 in using successive positive integers in place of successive prime numbers. [Modified by Stefano Spezia, Dec 20 2019 at the suggestion of Michel Marcus]
The trace of M(n) is A000027(n).
The sum of the first row of M(n) is A000217(n).
The sum of the first column of M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 19 2019]
For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - Stefano Spezia, Dec 29 2019
For n > 1 and k > 0, the sum of the k-th subdiagonal of M(n) is A120070(n,k). - Stefano Spezia, Dec 31 2019

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with permanent a(2) = 7.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with permanent a(3) = 100.

Stefano Spezia, Table of n, a(n) for n = 0..35
Wikipedia, Toeplitz Matrix

Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
         seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
seq(a(n), n = 0 .. 15);

b[n_]:=n; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]]; Array[a, 15,0]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 19 2019

a(0) = 1 prepended by Stefano Spezia, Dec 19 2019

A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 4, 3, 5, 6, 6, 4, 7, 8, 4, 5, 9, 9, 10, 6, 5, 12, 12, 5, 10, 14, 6, 8, 15, 8, 16, 6, 7, 18, 6, 9, 19, 20, 8, 7, 21, 10, 22, 12, 7, 24, 24, 7, 14, 15, 10, 14, 27, 12, 8, 9, 11, 30, 30, 8

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

			1 divides 2^2-1^2, so a(1)=2
2 divides 3^2-1^2, so a(2)=3
3 divides 2^2-a^2, so a(3)=2
4 divides 3^2-a^2, so a(4)=3

Cf. A000290, A204892.

s[n_] := s[n] = n^2; z1 = 600; z2 = 60;
Table[s[n], {n, 1, 30}]     (* A000290 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A120070 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]   (* A204994 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A204905 *)
Table[j[n], {n, 1, z2}]       (* A204995 *)
Table[s[k[n]], {n, 1, z2}]    (* A204996 *)
Table[s[j[n]], {n, 1, z2}]    (* A204997 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204998 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204999 *)

A143814 Triangle T(n,m) read along rows: T(n,m) = n^2 - (m+1)^2 for 1<=m

Original entry on oeis.org

3, 5, 8, 12, 7, 15, 21, 16, 9, 24, 32, 27, 20, 11, 35, 45, 40, 33, 24, 13, 48, 60, 55, 48, 39, 28, 15, 63, 77, 72, 65, 56, 45, 32, 17, 80, 96, 91, 84, 75, 64, 51, 36, 19, 99, 117, 112, 105, 96, 85, 72, 57, 40, 21, 120, 140, 135, 128, 119, 108, 95, 80, 63, 44, 23, 143

Offset: 2

Paul Curtz, Sep 02 2008

The triangle appears taking the entries of A140978,
4;
9,9;
16,16,16;
25,25,25,25;
..
minus the entries of A133819 with the 1's moved to the end of the rows,
1;
4,1;
4,9,1;
4,9,16,1;
4,9,16,25,1;
The result T(n,m) is a variant of A120070, the first term in each row of A120070 transferred to the end of the row.

			3;
5,8;
12,7,15;
21,16,9,24;
32,27,20,11,35;

Cf. A016061 (row sums).

A143814 := proc(n,m) if mA143814(n,m),m=1..n-1),n=2..14) ; # R. J. Mathar, Jan 23 2011

A261046 Irregular triangle read by rows: the first column consists of the odd numbers repeated but without the first 1. Row n (n>=0) contains floor(n/2)=1 terms. Every row contains successive odd numbers.

Original entry on oeis.org

1, 3, 3, 5, 5, 7, 5, 7, 9, 7, 9, 11, 7, 9, 11, 13, 9, 11, 13, 15, 9, 11, 13, 15, 17, 11, 13, 15, 17, 19, 11, 13, 15, 17, 19, 21, 13, 15, 17, 19, 21, 23, 13, 15, 17, 19, 21, 23, 25, 15, 17, 19, 21, 23, 25, 27, 13, 15, 17, 19, 21, 23, 25, 27

Offset: 0

Paul Curtz, Nov 19 2015

A131507(n), not in the same order.
a(n) multiplied by the triangle (extended A249947(n+1)) = (A167268(n+1))/2 is
   1,                 1,               1,
   3,                 1,               3,
   3, 5,              3, 1,            9, 5,
   5, 7,          *   3, 1,       =   15, 7,
   5, 7,  9,          5, 3, 1,        25, 21, 9
   7, 9, 11,          5, 3, 1,        35, 27, 11,
   etc.               etc.            etc.
The latter triangle is the odd numbers of A094728(n+1) which is
   1,
   4,  3,
   9,  8,  5,
  16, 15, 12,  7,
  25, 24, 21, 16, 9,
  etc.
Without the first column, the triangle is A120070(n+2). This gives a link between the frequencies of the spectral lines of the H-atom and the Janet periodic table of the elements.

			Triangle begins:
1,
3,
3,  5,
5,  7,
5,  7,  9,
7,  9, 11,
7,  9, 11, 13,
9, 11, 13, 15,
9, 11, 13, 15, 17,
....

Cf. A094728, A120070, A131507, A167268, A249947.

A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

Original entry on oeis.org

1, 3, 9, 5, 15, 7, 25, 21, 9, 35, 27, 11, 49, 45, 33, 13, 63, 55, 39, 15, 81, 77, 65, 45, 17, 99, 91, 75, 51, 19, 121, 117, 105, 85, 57, 21, 143, 135, 119, 95, 63, 23, 169, 165, 153, 133, 105, 69, 25, 195, 187, 171, 147, 115, 75, 27, 225, 221, 209, 189, 161, 125, 81, 29, 255, 247

Offset: 0

Paul Curtz, Nov 25 2015

A094728(n+1) comes from A120070(n+2). a(n) approximates frequencies of the spectral lines of the hydrogen atom.
Row sums: 1, 3, 14, 22, ... = A024598(n+1).
First column: A085046(n+1).
Row sums of A261046(n) = 1, 3, 8, 12, ... = A014255(n). See the formula.

			Irregular triangle begins:
1,
3,
9,  5,
15, 7,
25, 21,  9,
35, 27, 11,
49, 45, 33, 13,
63, 55, 39, 15,
...

Cf. A014255, A024598, A085046, A094728, A120070, A167268, A249947, A261046.

Table[n^2 - k^2, {n, 14}, {k, 0, n - 1}] /. n_ /; EvenQ@ n -> Nothing // Flatten (* Michael De Vlieger, Nov 25 2015 *)

for(n=1,20,for(k=0,n-1,s=n^2-k^2;if(s%2,print1(s,", ")))) \\ Derek Orr, Dec 24 2015

a(n) = A261046(n)*A167268(n+1)/2, where A167268 is Janet's sequence.

A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

Original entry on oeis.org

1, 3, 3, 6, 6, 5, 10, 10, 9, 7, 15, 15, 14, 12, 9, 21, 21, 20, 18, 15, 11, 28, 28, 27, 25, 22, 18, 13, 36, 36, 35, 33, 30, 26, 21, 15, 45, 45, 44, 42, 39, 35, 30, 24, 17, 55, 55, 54, 52, 49, 45, 40, 34, 27, 19, 66, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21

Offset: 0

Paul Curtz, Apr 12 2016

Row sums: A084990(n+1).
A158405(n) = A002262(n) + A002260(n). See the formula.
(Without its first column, A094728 is A120070, which could be built from positive A005563 and -A158894.)

			a(0) = 1, a(1) = 3, a(2) =3-0 = 3,  a(3) = 6, a(4) =6-0= 6, a(5) =6-1= 5, ... .
Triangle:
1,
3,   3,
6,   6,  5,
10, 10,  9,  7,
15, 15, 14, 12,  9,
21, 21, 20, 18, 15, 11,
28, 28, 27, 25, 22, 18, 13,
36, 36, 35, 33, 30, 26, 21, 15,
etc.

Cf. A000096, A000217, A002260, A002262, A005408, A005563, A049780, A084990, A094728, A120070, A158405, A158894.

Table[(n^2 - n)/2 - Prepend[Accumulate@ Range[0, n - 3], 0], {n, 12}] // Flatten (* Michael De Vlieger, Apr 12 2016 *)

a(n) = A094728(n+1) - A049780(n).

A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.

Original entry on oeis.org

1, 4, 5, 10, 16, 14, 20, 35, 40, 30, 35, 64, 81, 80, 55, 56, 105, 140, 154, 140, 91, 84, 160, 220, 256, 260, 224, 140, 120, 231, 324, 390, 420, 405, 336, 204, 165, 320, 455, 560, 625, 640, 595, 480, 285, 220, 429, 616, 770, 880, 935, 924, 836, 660, 385, 286, 560, 810, 1024

Offset: 2

Ali Sada and Yifan Xie, Jun 14 2022

Given a Pythagorean triple (a,b,c), define S = c^4 - a^4 - b^4. Using Euclid's parameterization (a = 2*n*k, b = n^2 - k^2, c = n^2 + k^2), substituting to get S in terms of n and k gives S = 8*n^2*k^2*((n^2 - k^2))^2, which is a multiple of 288; T(n, k) = sqrt(S/288) = n*k*(n^2 - k^2)/6 = n*k*(n+k)*(n-k)/6.

			Triangle begins:
  n/k   1    2    3    4    5    6    7
  2     1;
  3     4,   5;
  4    10,  16,  14;
  5    20,  35,  40,  30;
  6    35,  64,  81,  80,  55;
  7    56, 105, 140, 154, 140,  91;
  8    84, 160, 220, 256, 260, 224, 140;
  ...
For n = 3, k = 2, a = 5, b = 12, c = 13. T(3, 2) = sqrt((13^4 - 5^4 - 12^4)/288) = 5.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 72.

M. F. Hasler, Table of n, a(n) for n = 2..1000, May 08 2025
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A120070 (b leg), A055096 (c hypotenuse).

Cf. A006414 (row sums), A000292 (column 1), A077414 (column 2), A000330 (diagonal), A107984 (transpose), A210440 (diagonal which begins with 4).

T[n_,k_]:=n*k(n^2-k^2)/6; Table[T[n,k],{n,2,11},{k,n-1}]//Flatten (* Stefano Spezia, Jul 11 2025 *)

apply( {A354968(n, k=0)=k|| k=n-1-(1-n=ceil(sqrt(8*n-7)/2+.5))*(2-n)\2; k*(n-k)*n*(n+k)\6}, [2..66]) \\ M. F. Hasler, May 08 2025

G.f.: x^2*y*(1 + x*y - 4*x^2*y + x^3*y + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 11 2025

A132169 Irregular triangle read by rows. A141616(n)/4.

Original entry on oeis.org

2, 3, 6, 4, 8, 5, 12, 10, 6, 15, 12, 7, 20, 18, 14, 8, 24, 21, 16, 9, 30, 28, 24, 18, 10, 35, 32, 27, 20, 11, 42, 40, 36, 30, 22, 12, 48, 45, 40, 33, 24, 13, 56, 54, 50, 44, 36, 26, 14, 63, 60, 55, 48, 39, 28, 15, 72, 70, 66, 60, 52, 42, 30, 16

Offset: 0

Paul Curtz, Aug 26 2008

From Paul Curtz, Apr 14 2016: (Start)
Row sums: A023856.
Even rows: A120070.
Odd rows:
2,
6,   4,
12, 10, 6,
etc.
Divided by 2:
1,
3,   2,
6,   5,  3,
10,  9,  7, 4,
15, 14, 12, 9, 5,
etc.
This is A049777. Or positive A049780.
Also A271668 without the first column and bordered by the natural numbers as main diagonal.
(End)

			Irregular triangle:
2,
3,
6,   4,
8,   5,
12, 10, 6,
15, 12, 7,
20, 18, 14,  8,
24, 21, 16,  9,
30, 28, 24, 18, 10,
35, 32, 27, 20, 11,
etc.

Cf. A023856, A049777, A049780, A052938, A120070, A131723, A141616, A168230, A198442, A271668.

(Table[n^2 - k^2, {n, 3, 18}, {k, n}] /. m_ /; Or[OddQ@ m, m == 0] -> Nothing)/4 // Flatten (* Michael De Vlieger, Apr 14 2016 *)

Edited by Charles R Greathouse IV, Nov 11 2009

cp's OEIS Frontend

A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

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A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

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A143814 Triangle T(n,m) read along rows: T(n,m) = n^2 - (m+1)^2 for 1<=m

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A261046 Irregular triangle read by rows: the first column consists of the odd numbers repeated but without the first 1. Row n (n>=0) contains floor(n/2)=1 terms. Every row contains successive odd numbers.

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A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

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A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

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A354968 Triangle read by rows: T(n, k) = n*k*(n+k)*(n-k)/6.

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A132169 Irregular triangle read by rows. A141616(n)/4.

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A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.