A188866 -id:A188866 - cp's OEIS frontend

A188864 Number of n X 7 binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

128, 577, 1220, 1931, 2658, 3387, 4116, 4845, 5574, 6303, 7032, 7761, 8490, 9219, 9948, 10677, 11406, 12135, 12864, 13593, 14322, 15051, 15780, 16509, 17238, 17967, 18696, 19425, 20154, 20883, 21612, 22341, 23070, 23799, 24528, 25257, 25986, 26715

Offset: 1

R. H. Hardin, Apr 12 2011

nonn

Column 7 of A188866.

			Some solutions for 3 X 7:
..1..1..1..1..1..1..1....1..1..1..1..1..1..1....1..1..1..1..1..1..1
..1..1..1..1..1..1..1....1..1..0..1..1..1..1....1..1..1..1..1..1..1
..1..1..1..1..1..1..0....0..0..0..0..0..1..1....1..1..0..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A188866.

Empirical: a(n) = 729*n - 987 for n>4.

Empirical g.f.: x*(128 + 321*x + 194*x^2 + 68*x^3 + 16*x^4 + 2*x^5) / (1 - x)^2. - Colin Barker, May 01 2018

A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board.

Original entry on oeis.org

1, 2, 4, 3, 7, 17, 4, 10, 26, 68, 5, 13, 35, 95, 259, 6, 16, 44, 122, 340, 950, 7, 19, 53, 149, 421, 1193, 3387, 8, 22, 62, 176, 502, 1436, 4116, 11814, 9, 25, 71, 203, 583, 1679, 4845, 14001, 40503, 10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946, 11, 31, 89, 257, 745, 2165, 6303, 18375, 53625, 156629, 457795

Offset: 1

R. J. Mathar, Dec 13 2017

			Triangle begins:
   1;
   2,  4;
   3,  7, 17;
   4, 10, 26,  68;
   5, 13, 35,  95, 259;
   6, 16, 44, 122, 340,  950;
   7, 19, 53, 149, 421, 1193, 3387;
   8, 22, 62, 176, 502, 1436, 4116, 11814;
   9, 25, 71, 203, 583, 1679, 4845, 14001, 40503;
  10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946;

D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I, arXiv:1612.08697 [math.CO], 2016-2017, array I(m,n).

Cf. A081113 (diagonal), A000079 (2nd row), A001333 (3rd row), A126358, A057960, A126360, A002714, A126362, A188866.

Inm := proc(n,m)
    if m >= n then
        (n+2)*3^(n-2)+(m-n)*add(A005773(i)*A005773(n-i),i=0..n-1)
            +2*add((n-k-2)*3^(n-k-3)*A001006(k),k=0..n-3) ;
    else
        0 ;
    end if;
end proc:
for m from 1 to 13 do
for n from 1 to m do
    printf("%a,",Inm(n,m)) ;
end do:
printf("\n") ;
end do:
# Second program:
A296449row := proc(n) local gf, ser;
gf := n -> 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 +
ChebyshevU(n - 1, (1 - x)/(2*x))) / ChebyshevU(n, (1 - x)/(2*x)))/(1 - 3*x)^2;
ser := n -> series(expand(gf(n)), x, n + 1);
seq(coeff(ser(n), x, k), k = 1..n) end:
for n from 0 to 11 do A296449row(n) od; # Peter Luschny, Sep 07 2021

(* b = A005773 *) b[0] = 1; b[n_] := Sum[k/n*Sum[Binomial[n, j] * Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
(* c = A001006 *) c[0] = 1; c[n_] := c[n] = c[n-1] + Sum[c[k] * c[n-2-k], {k, 0, n-2}];
Inm[n_, m_] := If[m >= n, (n + 2)*3^(n - 2) + (m - n)*Sum[b[i]*b[n - i], {i, 0, n - 1}] + 2*Sum[(n - k - 2)*3^(n - k - 3)*c[k], {k, 0, n-3}], 0];
Table[Inm[n, m], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 23 2018, adapted from first Maple program. *)

I(m,n) = (n+2)*3^(n-2) + (m-n)*Sum_{i=0..n-1} A005773(i)*A005773(n-i) + 2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corr. 2.10]

I(m,n) = A188866(m-1,n) for m > 1. - Pontus von Brömssen, Sep 06 2021

cp's OEIS Frontend

A188864 Number of n X 7 binary arrays without the pattern 0 1 diagonally, vertically or antidiagonally.

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