A008810 -id:A008810 - cp's OEIS frontend

A358161 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ceiling(i*j/3).

1, 1, 3, 19, 434, 18142, 1138592, 131646240, 22247821152, 4990553682336, 1661493079305216, 729074911776673536, 397903630707426852864, 290086114501734871449600, 262660633302518916820992000, 284075108357948520100761600000, 385808192325346588875691868160000, 626209817056857125529475382231040000

Offset: 0

Stefano Spezia, Nov 01 2022

nonn

The matrix M(n) is the n-th principal submatrix of the rectangular array A143977.

det(M(n)) = 1 for n <= 3, and otherwise det(M(n)) = 0.

			a(5) = 18142:
    1  1  1  2  2
    1  2  2  3  4
    1  2  3  4  5
    2  3  4  6  7
    2  4  5  7  9

Cf. A143977.

Cf. A008810 (matrix element M[n,n]), A070333 (trace of M(n+1)), A358162 (hafnian of M(2*n)).

a[n_]:=Permanent[Table[Ceiling[i j/3],{i,n},{j,n}]]; Join[{1},Array[a,17]]

from fractions import Fraction
from sympy import Matrix
def A358161(n): return Matrix(n,n,[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

A222170 a(n) = n^2 + 2*floor(n^2/3).

Original entry on oeis.org

0, 1, 6, 15, 26, 41, 60, 81, 106, 135, 166, 201, 240, 281, 326, 375, 426, 481, 540, 601, 666, 735, 806, 881, 960, 1041, 1126, 1215, 1306, 1401, 1500, 1601, 1706, 1815, 1926, 2041, 2160, 2281, 2406, 2535, 2666, 2801, 2940, 3081, 3226, 3375, 3526, 3681, 3840

Offset: 0

Bruno Berselli, Aug 08 2013

Also, a(n) = n^2 + floor(2*n^2/3), since 2*floor(n^2/3) = floor(2*n^2/3).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tadeusz Dorozinskis, Pentagonpolyhedra Doro
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Subsequence of A008851.

Cf. A004773 (numbers of the type n+floor(n/3)), A008810 (numbers of the type n^2-2*floor(n^2/3)), A047220 (numbers of the type n+floor(2*n/3)), A184637 (numbers of the type n^2+floor(n^2/3), except the first two).

Magma
```
[n^2+2*Floor(n^2/3): n in [0..50]];
```

I:=[0,1,6,15,26]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[n^2 + 2 Floor[n^2/3], {n, 0, 50}]
CoefficientList[Series[x (1 + x) (1 + 3 x + x^2) / ((1 + x + x^2) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 6, 15, 26}, 50] (* Hugo Pfoertner, Jan 17 2023 *)

G.f.: x*(1+x)*(1 + 3*x + x^2)/((1 + x + x^2)*(1-x)^3).
a(n) = a(-n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(n) = floor(5*n^2/3). - Wesley Ivan Hurt, Mar 16 2015
a(n) = a(n-3) + 5*(2n-3) [Tadeusz Dorozinski]. - Eduard Baumann, Jan 18 2023

A329583 Numerators of 1 + n^2/4 + period 3: repeat [-1, 1, 1].

Original entry on oeis.org

0, 6, 3, 12, 6, 30, 9, 54, 18, 84, 27, 126, 36, 174, 51, 228, 66, 294, 81, 366, 102, 444, 123, 534, 144, 630, 171, 732, 198, 846, 225, 966, 258, 1092, 291, 1230, 324, 1374, 363, 1524, 402, 1686, 441, 1854, 486, 2028, 531, 2214, 576, 2406, 627

Offset: 0

Paul Curtz, Nov 17 2019

First bisection is 3*A008810.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2,3,0,-3,-2,1,1).

Cf. A008810, A042965, A047395, A130823, A131561, A261327.

MapIndexed[#1 - 2 Boole[Mod[First@ #2, 3] == 1] + 1 &, CoefficientList[Series[(1 + 5 x - x^2 - 2 x^3 + 2 x^4 + 5 x^5)/(1 - x^2)^3, {x, 0, 44}], x]] (* Michael De Vlieger, Nov 18 2019 *)

concat(0, Vec(3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Nov 24 2019

a(n) = A261327(n) + A131561(n+2) = (n^2 + 4)*(5 - 3*(-1)^n)/8 + (-1)^((n+1) mod 3).
From Colin Barker, Nov 24 2019: (Start)
G.f.: 3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)).
a(n) = -a(n-1) + 2*a(n-2) + 3*a(n-3) - 3*a(n-5) - 2*a(n-6) + a(n-7) + a(n-8) for n>8. (End)

Incorrect 129 replaced with 123 by Colin Barker, Nov 24 2019

A358162 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ceiling(i*j/3).

Original entry on oeis.org

1, 1, 11, 530, 71196, 18680148, 8825763888, 6969574132560, 8223753750015600, 14043461354695317600, 33726601900489760438400

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143977.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143977.

Cf. A008810 (matrix element M[n,n]), A070333 (trace of M(n)), A358161 (permanent of M(n)).

M[i_, j_, n_]:=Part[Part[Table[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, 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

A365539 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 + x^k)/((1 - x)^2*(1 - x^k)), with k > 0.

Original entry on oeis.org

1, 4, 1, 9, 2, 1, 16, 5, 2, 1, 25, 8, 3, 2, 1, 36, 13, 6, 3, 2, 1, 49, 18, 9, 4, 3, 2, 1, 64, 25, 12, 7, 4, 3, 2, 1, 81, 32, 17, 10, 5, 4, 3, 2, 1, 100, 41, 22, 13, 8, 5, 4, 3, 2, 1, 121, 50, 27, 16, 11, 6, 5, 4, 3, 2, 1, 144, 61, 34, 21, 14, 9, 6, 5, 4, 3, 2, 1

Offset: 0

Stefano Spezia, Sep 08 2023

			Array begins:
   1,  1,  1,  1,  1,  1,  1, ...
   4,  2,  2,  2,  2,  2,  2, ...
   9,  5,  3,  3,  3,  3,  3, ...
  16,  8,  6,  4,  4,  4,  4, ...
  25, 13,  9,  7,  5,  5,  5, ...
  36, 18, 12, 10,  8,  6,  6, ...
  49, 25, 17, 13, 11,  9,  7, ...
  64, 32, 22, 16, 14, 12, 10, ...
  ...

Cf. A000027 (main diagonal and superdiagonals), A000290 (k=1), A000982 (k=2), A008810 (k=3), A008811 (k=4), A008812 (k=5), A008813 (k=6), A008814 (k=7), A008815 (k=8), A008816 (k=9), A008817 (k=10).

Cf. A365540 (antidiagonal sums).

A[n_,k_]:=SeriesCoefficient[(1+x^k)/((1-x)^2*(1-x^k)),{x,0,n}]; Table[A[n-k,k],{n,0,12},{k,n}]//Flatten

A378763 Lower matching number for the n X n torus grid graph.

Original entry on oeis.org

3, 6, 9, 12, 17, 22, 27, 34, 42, 48, 58, 67, 75, 87

Offset: 3

Eric W. Weisstein, Dec 06 2024

a(18) = 108.
Seems to be either A008810(n) = ceil(n^2/3) or A008810(n)+1.
For known terms, a(n) is one more than A008810(n) for n = 11, 13, 14, 16.

Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Torus Grid Graph.

Cf. A008810 (ceil(n^2/3)).

Cf. A280984 (lower matching number for the n X n grid graph).

Table[Min[Length /@ FindIndependentVertexSet[LineGraph @ GraphProduct[CycleGraph[n], CycleGraph[n], "Cartesian"], Infinity, All]], {n, 3, 5}]

a(n) = n^2/3 if 3|n, otherwise a(n) >= A008810(n). [Proof: let the [minimum] maximal independent edge set be E. Let b be the number of edges between two vertices incident to different edges from E. The total number of edges connecting a vertex incident to an edge from E and a vertex not incident to any edge from E is equal to 4(n^2-2a(n)) but also to 6a(n)-2b; equalising these, we find b = 7a(n)-2n^2. Also, a(n) <= b, which gives the desired inequality a(n) >= n^2/3.] - Andrey Zabolotskiy, Dec 19 2024

A343953 Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x(1+x^n)/((1-x)^2(1-x^n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 0, 1, 2, 9, 0, 1, 2, 5, 16, 0, 1, 2, 3, 8, 25, 0, 1, 2, 3, 6, 13, 36, 0, 1, 2, 3, 4, 9, 18, 49, 0, 1, 2, 3, 4, 7, 12, 25, 64, 0, 1, 2, 3, 4, 5, 10, 17, 32, 81, 0, 1, 2, 3, 4, 5, 8, 13, 22, 41, 100, 0, 1, 2, 3, 4, 5, 6, 11, 16, 27, 50, 121, 0, 1, 2, 3, 4, 5, 6, 9, 14, 21, 34, 61, 144

Offset: 0

Jean-François Alcover, May 05 2021

			Square array begins:
0, 1, 4, 9,16,25,36,49,64,81,100,121,   ... (A000290)
0, 1, 2, 5, 8,13,18,25,32,41, 50, 61,   ... (A000982)
0, 1, 2, 3, 6, 9,12,17,22,27, 34, 41,   ... (A008810)
0, 1, 2, 3, 4, 7,10,13,16,21, 26, 31,   ... (A008811)
0, 1, 2, 3, 4, 5, 8,11,14,17, 20, 25,   ... (A008812)
0, 1, 2, 3, 4, 5, 6, 9,12,15, 18, 21,   ... (A008813)
0, 1, 2, 3, 4, 5, 6, 7,10,13, 16, 19,   ... (A008814)
0, 1, 2, 3, 4, 5, 6, 7, 8,11, 14, 17,   ... (A008815)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15,   ... (A008816)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13,   ... (A008817)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14,... not in the OEIS
...

Cf. A000290, A000982, A008810, A008811, A008812, A008813, A008814, A008815, A008816, A008817.

nmax = 15;
ro[n_] := ro[n] = CoefficientList[x(1+x^n)/((1-x)^2 (1-x^n))+O[x]^nmax, x];
T[n_, k_] := ro[n][[k+1]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}]  // Flatten

G.f. of row n: x*(1+x^n)/((1-x)^2*(1-x^n)), some cross-referenced sequences omitting the factor x and the initial term 0.

A371154 Maximum number of vertices for a given diameter n of a Cayley digraph on the cyclic group with generators s=1 and t>s.

Original entry on oeis.org

1, 3, 5, 8, 11, 16, 21, 26, 33, 40, 47, 56, 65, 74, 85, 96, 107, 120, 133, 146, 161, 176, 191, 208, 225, 242, 261, 280, 299, 320, 341, 362, 385, 408, 431, 456, 481, 506, 533, 560, 587

Offset: 0

Miquel A. Fiol, Mar 13 2024

			For n=10, the maximum number of vertices a(n)=47 is obtained, for instance, with the Cayley digraph Cay(47;1,11).

M. A. Fiol, J. L. A. Yebra, I. Alegre, and M. Valero Discrete optimization problem in local networks and data alignment, IEEE Trans. Comput., C-36 (1987), no. 6, 702-713.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Essentially A008810 - 1.

CoefficientList[Series[(1 + x - x^4 + 2*x^5 - x^6)/((1 - x)^3*(1 + x + x^2)),{x,0,40}],x] (* or *) Join[{1,3},Table[Ceiling[(n+2)^2/3]-1, {n,2,40}]] (* James C. McMahon, Apr 04 2024 *)

a(n) = ceiling((n+2)^2/3)-1 for n<>1.

G.f.: (1 + x - x^4 + 2*x^5 - x^6)/((1 - x)^3*(1 + x + x^2)). - Stefano Spezia, Mar 13 2024

cp's OEIS Frontend

A358161 a(n) is the permanent of the n X n matrix M(n) that is defined by M[i,j] = ceiling(i*j/3).

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A222170 a(n) = n^2 + 2*floor(n^2/3).

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Magma

Magma

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A329583 Numerators of 1 + n^2/4 + period 3: repeat [-1, 1, 1].

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A358162 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ceiling(i*j/3).

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A365539 Array read by ascending antidiagonals: A(n,k) = [x^n] (1 + x^k)/((1 - x)^2*(1 - x^k)), with k > 0.

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A378763 Lower matching number for the n X n torus grid graph.

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A343953 Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x*(1+x^n)/((1-x)^2*(1-x^n)).

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A371154 Maximum number of vertices for a given diameter n of a Cayley digraph on the cyclic group with generators s=1 and t>s.

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A343953 Square array T(n,k), n>=1, k>=0, read by antidiagonals, where row n is the expansion of x(1+x^n)/((1-x)^2(1-x^n)).