A143976 -id:A143976 - cp's OEIS frontend

A143974 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark those having x+y=1(mod 3); then R(m,n) is the number of marked unit squares in the rectangle [0,m]x[0,n].

0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 3, 4, 4, 3, 2, 2, 4, 5, 5, 5, 4, 2, 2, 4, 6, 6, 6, 6, 4, 2, 3, 5, 7, 8, 8, 8, 7, 5, 3, 3, 6, 8, 9, 10, 10, 9, 8, 6, 3, 3, 6, 9, 10, 11, 12, 11, 10, 9, 6, 3, 4, 7, 10, 12, 13, 14, 14, 13, 12, 10, 7, 4, 4, 8, 11, 13, 15, 16, 16, 16, 15, 13, 11, 8, 4, 4, 8

Offset: 1

Clark Kimberling, Sep 06 2008

Diagonals: A000212, A128422, A032765, A143975, A071408.

Rows: A002264, A004523, A000027, A004773, A138235, A005843, A047349 et al.

			Northwest corner:
0 0 1 1 1 2
0 1 2 2 3 4
1 2 3 4 5 6
1 2 4 5 6 8
1 3 5 6 8 10
R(3,4) counts these marked squares: (1,3), (2,2), (3,1), (3,4).

Cf. A143976, A143977.

R(m,n)=floor(mn/3).

A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 3, 4, 5, 6, 5, 4, 3, 3, 5, 6, 7, 7, 6, 5, 3, 3, 6, 7, 8, 9, 8, 7, 6, 3, 4, 6, 8, 10, 10, 10, 10, 8, 6, 4, 4, 7, 9, 11, 12, 12, 12, 11, 9, 7, 4, 4, 8, 10, 12, 14, 14, 14, 14, 12, 10, 8, 4, 5, 8, 11, 14, 15, 16, 17, 16, 15, 14, 11, 8, 5

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...) = A000027.

			Northwest corner:
  1  1  1  2  2  2  3
  1  2  2  3  4  4  5
  1  2  3  4  5  6  7
  2  3  4  6  7  8 10
  2  4  5  7  9 10 12

Stefano Spezia, First 140 antidiagonals of the array, flattened

Diagonals: A008810, A007984, A000212, A128422.
Rows and columns: A002264, A004523, A000027, A004772, A047212, et al.
Cf. A143974, A143976, A143979.

T[m_,n_]:=Ceiling[m n/3];Flatten[Table[T[m-n+1,n],{m,13},{n,m}]] (* Stefano Spezia, Oct 27 2022 *)

R(m,n) = ceiling(m*n/3). [Corrected by Stefano Spezia, Oct 27 2022]

A143979 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| = 0 (mod 3); then R(m,n) is the number of UNmarked squares in the rectangle [0,m] X [0,n].

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 4, 4, 2, 3, 5, 6, 5, 3, 4, 6, 8, 8, 6, 4, 4, 8, 10, 10, 10, 8, 4, 5, 9, 12, 13, 13, 12, 9, 5, 6, 10, 14, 16, 16, 16, 14, 10, 6, 6, 12, 16, 18, 20, 20, 18, 16, 12, 6, 7, 13, 18, 21, 23, 24, 23, 21, 18, 13, 7, 8, 14, 20, 24, 26, 28, 28, 26, 24, 20, 14, 8

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...)=A000027.

			Northwest corner:
  0 1  2  2  3  4  4
  1 2  4  5  6  8  9
  2 4  6  8 10 12 14
  2 5  8 10 13 16 18
  3 6 10 13 16 20 23

Cf. A143974, A143976, A143977.

T[i_,j_]:=i*j-Ceiling[i*j/3]; Flatten[Table[T[m-n+1,n],{m,12},{n,m}]] (* Stefano Spezia, Oct 28 2022 *)

R(m,n) = m*n - ceiling(m*n/3). [Corrected by Stefano Spezia, Oct 28 2022]

A358042 Partial sums of A071619.

Original entry on oeis.org

0, 1, 4, 10, 21, 38, 62, 95, 138, 192, 259, 340, 436, 549, 680, 830, 1001, 1194, 1410, 1651, 1918, 2212, 2535, 2888, 3272, 3689, 4140, 4626, 5149, 5710, 6310, 6951, 7634, 8360, 9131, 9948, 10812, 11725, 12688, 13702, 14769, 15890, 17066, 18299, 19590, 20940, 22351

Offset: 0

Stefano Spezia, Oct 26 2022

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Partial sums of the main diagonal of A143976.
Cf. A042968 (2nd differences), A071619 (1st differences).
Cf. A005898, A049347.

LinearRecurrence[{3,-3,2,-3,3,-1},{0,1,4,10,21,38},47]

O.g.f.: x*(1 + x)*(1 + x^2)/((1 + x + x^2)*(1 - x)^4).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) for n > 5.
a(n) = (A005898(n) - A049347(n))/9.
E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(1 + 8*x + 9*x^2 + 2*x^3) - 3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/27.

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - floor(ij/3).

Original entry on oeis.org

1, 1, 7, 102, 4396, 374216, 49857920, 11344877568, 3879729283968, 1804571320405248, 1195546731955854336, 1058730877124859138048, 1184751018265831288602624, 1725335046543668616765112320, 3147123030650561978295975936000, 6934187745940804400441946931200000, 18840570649600136750602236509552640000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143976 and it is singular for n > 3.

			a(5) = 374216:
    1   2   2   3   4
    2   3   4   6   7
    2   4   6   8  10
    3   6   8  11  14
    4   7  10  14  17

Cf. A143976.

Cf. A071619 (matrix element M[n,n]), A358042 (trace of M(n)), A358160 (hafnian of M(2*n)).

Join[{1},Table[Permanent[Table[i*j-Floor[i*j/3],{i,n},{j,n}]],{n,17}]]

from sympy import Matrix
def A358159(n): return Matrix(n,n,[i*j-i*j//3 for i in range(1,n+1) for j in range(1,n+1)]).per() if n else 1 # Chai Wah Wu, Nov 02 2022

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).

Original entry on oeis.org

1, 2, 40, 3884, 1016376, 534983256, 510252517152, 802452895865280, 1901953775079849600, 6537796866589765507200, 31381746234057256630521600

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143976.

			a(2) = 40:
    1   2   2   3
    2   3   4   6
    2   4   6   8
    3   6   8  11

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143976.

Cf. A071619 (matrix element M[n,n]), A358159 (permanent of M(2*n)), A358042 (trace of M(n)).

M[i_, j_, n_]:=Part[Part[Table[r*c-Floor[r*c/3], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i*j - (i*j)\3);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

cp's OEIS Frontend

A143974 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark those having x+y=1(mod 3); then R(m,n) is the number of marked unit squares in the rectangle [0,m]x[0,n].

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A143979 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| = 0 (mod 3); then R(m,n) is the number of UNmarked squares in the rectangle [0,m] X [0,n].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A358042 Partial sums of A071619.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = i*j - floor(i*j/3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - floor(i*j/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A358159 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - floor(ij/3).

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).