A143979 -id:A143979 - cp's OEIS frontend

A143978 a(n) = floor(2n(n+1)/3).

1, 4, 8, 13, 20, 28, 37, 48, 60, 73, 88, 104, 121, 140, 160, 181, 204, 228, 253, 280, 308, 337, 368, 400, 433, 468, 504, 541, 580, 620, 661, 704, 748, 793, 840, 888, 937, 988, 1040, 1093, 1148, 1204, 1261, 1320, 1380, 1441, 1504, 1568, 1633, 1700, 1768, 1837

Offset: 1

Clark Kimberling, Sep 06 2008

Second diagonal of array A143979, which counts certain unit squares in a lattice. First diagonal: A030511.
Convolution of A042965 with A000012, convolution of A131534 with A000027, and convolution of A106510 with A000217. - L. Edson Jeffery, Jan 24 2015
From Miquel A. Fiol, Aug 31 2024: (Start)
a(n+1) is the maximum number N of vertices of a circulant digraph with steps +-s1, s2, and diameter n.
Depending on the value of n, the following table shows the values of N, s1, and s2:
 n |     3*r     |   3*r-1   |   3*r-2   |
 N | 6*r^2+6*r+1 | 6*r^2+2*r | 6*r^2-2*r |
s1 |      1      |     r     |     r     |
s2 |    6*r+3    |   3*r+1   |   3*r-1   |
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Bruno Berselli, Illustration of the initial terms.
P. Morillo and Miquel A. Fiol, El diámetro de ciertos digrafos circulantes de triple paso, Stochastica X (1986), no. 3, 233-249.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000217, A030511, A042965 (first differences), A106510, A131534, A143979.

Cf. A049347, A102283, A103115, A131713.

A143978:= n-> (6*n*(n+1) -1 + `mod`(n+2,3) - `mod`(n+1,3))/9;
seq(A143978(n), n=1..60); # G. C. Greubel, May 27 2020

Table[(6*n^2 +6*n -1  + Mod[n+2, 3] - Mod[n+1, 3])/9, {n, 60}] (* G. C. Greubel, May 27 2020 *)
Table[Floor[2n (n+1)/3],{n,60}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,4,8,13,20},60] (* Harvey P. Dale, Aug 12 2025 *)

From R. J. Mathar, Oct 05 2009: (Start)
G.f.: x*(1 + x)^2/((1 + x + x^2)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). (End)
a(n) = Sum_{k=1..(n+1)} A042965(k). - Klaus Purath, May 23 2020
From G. C. Greubel, May 27 2020: (Start)
a(n) = (ChebyshevU(n, -1/2) - ChebyshevU(n-1, -1/2) + (6*n^2 + 6*n -1))/9.
a(n) = (JacobiSymbol(n+1, 3) - JacobiSymbol(n, 3) + (6*n^2 + 6*n -1))/9.
a(n) = (A102283(n+1) - A102283(n) + A103115(n+1))/9
a(n) = (A131713(n) + A103115(n+1))/9. (End)
Sum_{n>=1} 1/a(n) = 3/2 + (tan(Pi/(2*sqrt(3)))-1)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022
E.g.f.: exp(-x/2)*(exp(3*x/2)*(6*x^2 + 12*x - 1) + cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Apr 05 2023

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

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  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.

Stefano Spezia, First 140 antidiagonals of the array, flattened

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

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 *)

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

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]

A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - ceiling(ij/3).

Original entry on oeis.org

1, 0, 1, 30, 1272, 113224, 18615680, 4299553536, 1507609286784, 781464165813504, 525599814806986752, 473934337123421786112, 567876971785035135320064, 837723761443461191423754240, 1549608938859438129393893376000, 3582000047767392376356107059200000, 9838495669776145718724862743674880000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

det(M(0)) = 1, det(M(2)) = -1, det(M(3)) = 2, and otherwise det(M(n)) = 0.

			a(5) = 113224:
    0  1   2   2   3
    1  2   4   5   6
    2  4   6   8  10
    2  5   8  10  13
    3  6  10  13  16

Cf. A143979.

Cf. A030511 (matrix element M[n-1,n-1]), A358164 (hafnian of M(2*n)).

a[n_]:=Permanent[Table[i*j-Ceiling[i*j/3],{i,n},{j,n}]]; Join[{1},Array[a,16]]

from fractions import Fraction
from sympy import Matrix
def A358163(n): return Matrix(n,n,[i*j-Fraction(i*j,3)._ceil_() 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

A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - ceiling(ij/3).

Original entry on oeis.org

1, 1, 26, 2704, 698568, 384890688, 378771904512, 597991783196160, 1450380828625459200, 5077825865646165964800, 24487520383436615392204800

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143979.

			a(2) = 26:
    0  1   2   2
    1  2   4   5
    2  4   6   8
    2  5   8  10

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143979.

Cf. A030511 (matrix element M[n-1,n-1]), A358163 (permanent of M(n)).

M[i_, j_, n_]:=Part[Part[Table[r*c-Ceiling[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 - ceil((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

A143978 a(n) = floor(2*n*(n+1)/3).

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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].

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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].

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A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = i*j - ceiling(i*j/3).

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A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - ceiling(i*j/3).

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A143978 a(n) = floor(2n(n+1)/3).

A358163 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ij - ceiling(ij/3).

A358164 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - ceiling(ij/3).