cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A143976 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 == 1 (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

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 4, 6, 6, 6, 4, 4, 7, 8, 8, 7, 4, 5, 8, 10, 11, 10, 8, 5, 6, 10, 12, 14, 14, 12, 10, 6, 6, 11, 14, 16, 17, 16, 14, 11, 6, 7, 12, 16, 19, 20, 20, 19, 16, 12, 7, 8, 14, 18, 22, 24, 24, 24, 22, 18, 14, 8, 8, 15, 20, 24, 27, 28, 28, 27, 24, 20, 15, 8
Offset: 1

Views

Author

Clark Kimberling, Sep 06 2008

Keywords

Comments

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

Examples

			Northwest corner:
  1  2  2  3  4  4  5
  2  3  4  6  7  8 10
  2  4  6  8 10 12 14
  3  6  8 11 14 16 18
  4  7 10 14 17 20 24
See A143974.

Links

Crossrefs

Rows: A004523, A004772, A005843, A047399, et al.
Main diagonal: A071619.
Cf. A143974, A143977, A143979.

Programs

  • Mathematica
    T[m_,n_]:=m*n-Floor[m*n/3]; Flatten[Table[T[n-k+1,k],{n,12},{k,n}]] (* Stefano Spezia, Oct 25 2022 *)

Formula

R(m,n) = m*n - floor(m*n/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

Views

Author

Clark Kimberling, Sep 06 2008

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Clark Kimberling, Sep 06 2008

Keywords

Comments

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

Examples

			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

Crossrefs

Cf. A143974, A143976, A143977.

Programs

  • Mathematica
    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 *)

Formula

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

A143975 a(n) = floor(n*(n+3)/3).

Original entry on oeis.org

1, 3, 6, 9, 13, 18, 23, 29, 36, 43, 51, 60, 69, 79, 90, 101, 113, 126, 139, 153, 168, 183, 199, 216, 233, 251, 270, 289, 309, 330, 351, 373, 396, 419, 443, 468, 493, 519, 546, 573, 601, 630, 659, 689, 720, 751, 783, 816, 849, 883, 918, 953, 989, 1026, 1063, 1101
Offset: 1

Views

Author

Clark Kimberling, Sep 06 2008

Keywords

Comments

Fourth diagonal of A143974, associated with counting unit squares in a lattice.

Examples

			Main diagonal of A143974: (0,1,3,5,8,12,...) = A000212;
2nd diagonal: (0,2,4,6,10,14,18,...) = A128422;
3rd diagonal: (1,2,5,8,11,16,21,...) = A032765;
4th diagonal: (1,3,6,9,13,18,23,...) = A143975.

Links

Crossrefs

Cf. A099837, A143974.

Programs

  • Magma
    [Floor(n*(n+3)/3): n in [1..60]]; // Vincenzo Librandi, May 08 2011
  • Mathematica
    a[n_] := Floor[n*(n+3)/3]; Array[a, 60] (* Amiram Eldar, Oct 01 2022 *)

Formula

a(n) = floor(n*(n+3)/3).
From R. J. Mathar, Oct 05 2009: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
G.f.: x*(-1 - x - x^2 + x^3)/( (1 + x + x^2) * (x-1)^3). (End)
9*a(n) = 3*n^2 + 9*n - 2 + A099837(n+3). - R. J. Mathar, Apr 26 2022
Sum_{n>=1} 1/a(n) = 4/3 + (tan((sqrt(13)+2)*Pi/6) - cot((sqrt(13)+1)*Pi/6)) * Pi/sqrt(13). - Amiram Eldar, Oct 01 2022
E.g.f.: (exp(x)*(3*x*(4 + x) - 2) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022

A358157 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = floor(i*j/3).

Original entry on oeis.org

1, 0, 0, 1, 32, 1422, 146720, 18258864, 3217515264, 910849979232, 316878962588928, 143616562358849280, 90359341652805156864, 68004478547050644357120, 63187026071337208000512000, 75392341069747600992153600000, 104962910849766568886449582080000, 174017685915978467201007058206720000
Offset: 0

Views

Author

Stefano Spezia, Nov 01 2022

Keywords

Comments

The matrix M(n) is the n-th principal submatrix of the rectangular array A143974.
det(M(0)) = 1, det(M(3)) = -1, and otherwise det(M(n)) = 0.

Examples

			a(5) = 1422:
    0  0  1  1  1
    0  1  2  2  3
    1  2  3  4  5
    1  2  4  5  6
    1  3  5  6  8

Crossrefs

Cf. A143974.
Cf. A000212 (matrix element M[n,n]), A181286 (trace of M(n)), A358158 (hafnian of M(2*n)).

Programs

  • Mathematica
    a[n_]:=Permanent[Table[Floor[i j/3],{i,n},{j,n}]]; Join[{1},Array[a,17]]
  • Python
    from sympy import Matrix
def A358157(n): return Matrix(n,n,[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

A358158 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = floor(i*j/3).

Original entry on oeis.org

1, 0, 4, 238, 31992, 9390096, 4755878928, 3802500283680, 4720879431568800, 8379987002639042400, 20346893722025317036800
Offset: 0

Views

Author

Stefano Spezia, Nov 01 2022

Keywords

Comments

The matrix M(n) is the n-th principal submatrix of the rectangular array A143974.

Examples

			a(2) = 4:
    0  0  1  1
    0  1  2  2
    1  2  3  4
    1  2  4  5

Links

Crossrefs

Cf. A143974.
Cf. A000212 (matrix element M[n,n]), A181286 (trace of M(n)), A358157 (permanent of M(n)).

Programs

  • Mathematica
    M[i_, j_, n_]:=Part[Part[Table[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]
  • PARI
    tm(n) = matrix(n, n, 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

Extensions

a(6) from Michel Marcus, May 02 2023
a(7)-a(10) from Pontus von Brömssen, Oct 15 2023
Showing 1-6 of 6 results.

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