A351153 -id:A351153 - cp's OEIS frontend

A351154 a(n) is the permanent of the n X n matrix M(n) that is defined as M[i,j,n] = A351153(n, min(i, j)) + abs(i - j).

1, 1, 7, 169, 10388, 1324344, 305668180, 116145817656, 67770421715800, 57594670663866124, 68393751368082128320, 109765035421144948709232, 231657098706747226470685920, 628412716450312334529486247152, 2149132484027947970192241804640128, 9113755489596517688997731211571700256

Offset: 0

Stefano Spezia, Feb 02 2022

nonn

Conjectures: (Start)
det(M(0)) = det(M(1)) = 1 and det(M(n)) = -(n - 2)! for n > 1.
abs(det(M(n))) = abs(A159333(n-2)). (End)

			a(3) = 169:
    1    2    3
    2    4    5
    3    5    6
a(4) = 10388:
    1    2    3    4
    2    5    6    7
    3    6    8    9
    4    7    9   10

Cf. A000142, A003983, A049581, A159333, A351153.

A351153[n_,k_]:=n(k-1)-k(k-3)/2; M[i_,j_,n_]:=A351153[n,Min[i,j]]+Abs[i-j]; a[n_]:=Permanent[Table[M[i,j,n],{i,n},{j,n}]]; Join[{1},Array[a,15]]

t(n, k) = n*(k-1) - k*(k-3)/2; \\ A351153
a(n) = matpermanent(matrix(n, n, i, j, t(n, min(i, j)) + abs(i - j))); \\ Michel Marcus, Feb 03 2022

A374021 Row products of A351153.

Original entry on oeis.org

1, 1, 3, 24, 400, 11700, 536256, 35631232, 3245577984, 388702800000, 59265098200000, 11212953038217216, 2578459154484215808, 708372581870426497024, 229171184991141120000000, 86242576440372042240000000, 37355382389967084527220883456, 18452600861204793901808906698752

Offset: 0

Stefano Spezia, Jun 25 2024

nonn

Cf. A005408, A033996, A351153.

A351153[n_,k_]:=n(k-1)-k(k-3)/2; a[n_]:=Product[A351153[n,k],{k,n}]; Array[a,18,0]

a(n) = vecprod(vector(n, k, n*(k - 1) - k*(k - 3)/2)); \\ Michel Marcus, Jun 25 2024

a(n) = (-1/2)^n*Pochhammer((b(n)-2*n-1)/2, n)*Pochhammer((-b(n)-2*n-1)/2, n), where b(n) = sqrt(A033996(n) + 9).

a(n) ~ sqrt(Pi) * 2^(n+2) * n^(2*n - 3/2) / exp(2*n). - Vaclav Kotesovec, Jun 27 2024

A136107 Number of representations of n as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 1, 2, 4, 2, 1, 4, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 5, 2, 2, 2, 3, 3, 4, 2, 2, 4, 3, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 3, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 3, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 3, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 7

Offset: 1

John W. Layman and Robert G. Wilson v, Dec 12 2007

nonn

a(n) is also the number of partitions of n into consecutive parts greater than 1. - Omar E. Pol, Feb 07 2022
a(n) is the number of solutions of the equations 2(x-1)y-(x-3)x=2(n+1) for 0A351284; y-values in A351285. Also the number of times n+1 appears in A351153. - Stefano Spezia, Feb 12 2022
Equivalence with Stefano Spezia solutions: The equation 2(x-1)y-(x-3)x=2(n+1) can be rewritten (y+1-x/2)(x+1)=n; proof by solving both for y. So solutions factorize n, and since x+1 must be an integer and y+1-x/2 must be an integer, x must be even. So (x+1)|n means we are looking for odd divisors of n, which is the A001227 term of the Alekseyev formula. The correction by A010054 in the Alekseyev formula means: if n is a triangular number, the solution x=y+1 where x>y is not counted by Spezia. - R. J. Mathar, Feb 12 2022

			a(2) = 1 because 3 - 1 = 2,
a(5) = 2 because 6 - 1 = 15 - 10 = 5,
a(9) = 3 because 10 - 1 = 15 - 6 = 45 - 36 = 9, etc.
For n = 21 the four partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6] and [6, 5, 4, 3, 2, 1]. The last partition contains 1 as a part, hence there are only three partitions of 21 into consecutive parts whose parts are greater than 1, so a(21) = 3. - _Omar E. Pol_, Feb 07 2022

Robert G. Wilson v, Table of n, a(n) for n = 1..54000.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
Robert Dougherty-Bliss and Natalya Ter-Saakov, The Comma Sequence is Finite in Other Bases, arXiv:2408.03434 [math.NT], 2024.

Cf. A000217, A001227, A010054, A136108, A351153, A351284, A351285.

f[n_] := Block[{c = 0, k = 1}, While[k < n, If[ IntegerQ[ Sqrt[8 n + 4 k (k + 1) + 1]], c++ ]; k++ ]; c]; Table[f@n, {n, 105}]

a(n) = numdiv(n>>valuation(n, 2)) - ispolygonal(n, 3); \\ Michel Marcus, Jan 08 2024

G.f.: Sum_{n>=1} x^((n^2+3*n)/2)/(1-x^n). - Vladeta Jovovic, May 13 2008

a(n) = A001227(n) - A010054(n). - Max Alekseyev, May 13 2009

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 152, 7113, 745285, 94974369

Offset: 0

Stefano Spezia, Feb 14 2022

Upper bounds for the next terms can be found by considering all possibilities of choosing matrix entries on the diagonal and applying Gasper's determinant theorem (see references in A085000): a(7) <= 22475584128, a(8) <= 6634478203404, a(9) <= 2647044512044258. - Hugo Pfoertner, Feb 18 2022

			a(3) = 152:
   2    4    6
   4    5    1
   6    1    3
a(4) = 7113:
   2    6    8    9
   6    5   10    1
   8   10    3    4
   9    1    4    7

Cf. A000217, A351147, A351148, A351153.

Cf. A085000, A180128, A350931, A350933, A350954, A350956.

a(n) = max(abs(A351147(n)), A351148(n)). - Hugo Pfoertner, Feb 16 2022

a(5)-a(6) from Hugo Pfoertner, Feb 16 2022

A358806 a(n) is the minimal determinant of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Original entry on oeis.org

1, 0, -4, -110, -5072, -488212, -86577891

Offset: 0

Stefano Spezia, Dec 02 2022

			a(2) = -4:
    [0, 2;
     2, 1]
a(3) = -110:
    [1, 3, 5;
     3, 4, 0;
     5, 0, 2]

Cf. A000217, A351147, A351153.

Cf. A358807 (maximal), A358808 (minimal permanent), A358809 (maximal permanent).

A358807 a(n) is the maximal determinant of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Original entry on oeis.org

1, 0, 2, 86, 5911, 652189, 82173814

Offset: 0

Stefano Spezia, Dec 02 2022

			a(2) = 2:
    [1, 0;
     0, 2]
a(3) = 86:
    [0, 3, 4;
     3, 1, 5;
     4, 5, 2]

Cf. A000217, A351148, A351153.

Cf. A358806 (minimal), A358808 (minimal permanent), A358809 (maximal permanent).

A358808 a(n) is the minimal permanent of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Original entry on oeis.org

1, 0, 1, 33, 2425, 357046, 92052610

Offset: 0

Stefano Spezia, Dec 02 2022

			a(2) = 1:
    [0, 1;
     1, 2]
a(3) = 33:
    [0, 1, 2;
     1, 3, 4;
     2, 4, 5]
a(4) = 2425:
  [0, 2, 3, 1;
   2, 8, 7, 4;
   3, 7, 9, 6;
   1, 4, 6, 5]

Cf. A000217, A351153, A351610.

Cf. A358806 (minimal determinant), A358807 (maximal determinant), A358809 (maximal).

a(4) corrected and a(5)-a(6) from Hugo Pfoertner, Dec 07 2022

A358809 a(n) is the maximal permanent of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Original entry on oeis.org

1, 0, 4, 186, 21823, 4569098, 1713573909

Offset: 0

Stefano Spezia, Dec 02 2022

			a(2) = 4:
    [0, 2;
     2, 1]
a(3) = 186:
    [0, 4, 5;
     4, 2, 3;
     5, 3, 1]

Cf. A000217, A351153, A351611.

Cf. A358806 (minimal determinant), A358807 (maximal determinant), A358808 (minimal).

a(5)-a(6) from Hugo Pfoertner, Dec 07 2022

A351284 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 4, 2, 2, 3, 2, 4, 5, 2, 3, 2, 6, 2, 3, 4, 2, 5, 2, 3, 2, 4, 2, 3, 6, 2, 5, 2, 3, 4, 7, 2, 2, 3, 2, 4, 5, 6, 2, 3, 2, 2, 3, 4, 7, 2, 5, 2, 3, 6, 8, 2, 4, 2, 3, 2, 5, 2, 3, 4, 7, 2, 6, 2, 3, 2, 4, 5, 8

Offset: 3

Stefano Spezia, Feb 06 2022

Equivalently, the n-th row gives the column indices corresponding to n in the triangle A351153.

			Triangle begins:
  2;
  2;
  2;
  2, 3;
  2;
  2, 3;
  2;
  2, 3, 4;
  ...

Cf. A341829, A351153, A136107 (row length or solutions number), A351285 (y-values).

Mathematica
```
Table[r={};For[d=1,d
				
```

A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.

Original entry on oeis.org

2, 3, 4, 5, 3, 6, 7, 4, 8, 9, 5, 4, 10, 11, 6, 12, 5, 13, 7, 14, 5, 15, 8, 6, 16, 17, 9, 18, 7, 6, 19, 10, 20, 6, 21, 11, 8, 22, 7, 23, 12, 24, 9, 25, 13, 7, 26, 8, 27, 14, 10, 7, 28, 29, 15, 30, 11, 9, 8, 31, 16, 32, 33, 17, 12, 8, 34, 10, 35, 18, 9, 8, 36, 13

Offset: 3

Stefano Spezia, Feb 06 2022

Equivalently, the n-th row gives the row indices corresponding to n in the triangle A351153.

			Triangle begins:
  2;
  3;
  4;
  5, 3;
  6;
  7, 4;
  8;
  9, 5, 4;
  ...

Cf. A341830, A351153, A136107(row length or solutions number), A351284 (x-values).

Mathematica
```
Table[r={};For[d=1,d
				
```

cp's OEIS Frontend

A351154 a(n) is the permanent of the n X n matrix M(n) that is defined as M[i,j,n] = A351153(n, min(i, j)) + abs(i - j).

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A374021 Row products of A351153.

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A136107 Number of representations of n as the difference of two positive triangular numbers.

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A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

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A358806 a(n) is the minimal determinant of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

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A358807 a(n) is the maximal determinant of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

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Author

Keywords

Examples

Crossrefs

A358808 a(n) is the minimal permanent of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A358809 a(n) is the maximal permanent of an n X n symmetric matrix using all the integers from 0 to n*(n + 1)/2 - 1.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A351284 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation 2*(x - 1)*y - (x - 3)*x = 2*n for 0 < x <= y.

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Author

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Comments

Examples

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Programs

Mathematica

A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2*(x - 1)*y - (x - 3)*x = 2*n for 0 < x <= y.

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Author

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A351284 Irregular triangle read by rows: the n-th row gives the x-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.

A351285 Irregular triangle read by rows in which row n gives the y-values of the solutions of the equation 2(x - 1)y - (x - 3)x = 2n for 0 < x <= y.