A204623 -id:A204623 - cp's OEIS frontend

A208258 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than one of its immediate leftward or upward neighbors.

1, 2, 2, 5, 12, 5, 14, 96, 96, 14, 41, 768, 2040, 768, 41, 122, 6144, 43344, 43344, 6144, 122, 365, 49152, 920928, 2445264, 920928, 49152, 365, 1094, 393216, 19566912, 137947632, 137947632, 19566912, 393216, 1094, 3281, 3145728, 415737216

Offset: 1

R. H. Hardin Feb 24 2012

Table starts
....1.......2..........5.............14................41...................122
....2......12.........96............768..............6144.................49152
....5......96.......2040..........43344............920928..............19566912
...14.....768......43344........2445264.........137947632............7782199824
...41....6144.....920928......137947632.......20662697664.........3094987890864
..122...49152...19566912.....7782199824.....3094987890864......1230875651548608
..365..393216..415737216...439026252912...463586569433280....489518751475118880
.1094.3145728.8833148160.24767296501776.69438884409258336.194681408759751734640

			Some solutions for n=4 k=3
..0..0..0....0..1..0....0..0..0....0..1..2....0..0..1....0..0..1....0..0..1
..1..1..1....2..1..2....0..1..2....1..2..1....2..0..2....0..1..2....2..0..2
..1..0..0....2..1..1....2..2..1....0..1..2....1..0..2....1..2..0....2..0..2
..1..2..2....0..1..2....2..1..2....2..1..2....1..2..1....0..0..1....2..0..1

R. H. Hardin, Table of n, a(n) for n = 1..364

Column 1 is A007051(n-1)

Column 2 is A204623(n-1)

A270568 Expansion of g.f. (1+4x)/(1-8x).

Original entry on oeis.org

1, 12, 96, 768, 6144, 49152, 393216, 3145728, 25165824, 201326592, 1610612736, 12884901888, 103079215104, 824633720832, 6597069766656, 52776558133248, 422212465065984, 3377699720527872, 27021597764222976, 216172782113783808, 1729382256910270464, 13835058055282163712

Offset: 0

Colin Barker, Mar 19 2016

Partial sums are 1, 13, 109, 877, 7021, 56173, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A001018 (powers of 8), A094085, A204623.

CoefficientList[Series[(1 + 4 x)/(1 - 8 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 19 2016 *)
Join[{1},NestList[8#&,12,30]] (* Harvey P. Dale, Oct 05 2022 *)

PARI
```
Vec((1+4*x)/(1-8*x) + O(x^30))
```

G.f.: (1+4*x)/(1-8*x).
a(n) = 8*a(n-1) for n>1.
a(n) = 12*8^(n-1) for n>0.
a(n) = A094085(n), n>1. - R. J. Mathar, Mar 21 2016
E.g.f.: (3*exp(8*x) - 1)/2. - Elmo R. Oliveira, Mar 25 2025

A363099 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).

Original entry on oeis.org

11, 3, 12, 13, 91, 27, 96, 107, 3, 108, 109, 731, 219, 768, 859, 27, 864, 875, 3, 876, 877, 5851, 1755, 6144, 6875, 219, 6912, 7003, 27, 7008, 7019, 3, 7020, 7021, 46811, 14043, 49152, 55003, 1755, 55296, 56027, 219, 56064, 56155, 27, 56160, 56171, 3, 56172, 56173, 374491, 112347, 393216, 440027, 14043, 442368

Offset: 1

Nei Y. Soma, May 20 2023

Each term in the sequence encodes a line of the inverse of a Jacobi matrix that has 1s on its lower, main, and upper diagonals in GF(2). The associated inverse matrix column values come from the binary representation of that base-10 number, being a bit per column. These matrices start with a 4 X 4 matrix and the consecutive terms came by adding ascending and consecutive multiples of 3. If the binary number has fewer bits than the number of columns, it must be zero-padded to the left. To obtain the inverse matrices in real numbers instead of GF(2), alternate between + and - between the 1s in a row. If a row is a multiple of 3, alternate between - and +. The determinants of these 3m+1 X 3m+1 Jacobi matrices are 1 in GF(2), and alternate between -1 and 1 in R if m is odd or even. These properties were proven by Sutner (1989) and Melo (1997), respectively.

			For m = 1, the Jacobi 4 X 4 matrix has as rows
     1, 1, 0, 0
     1, 1, 1, 0
     0, 1, 1, 1
     0, 0, 1, 1
Its inverse has the rows
     1, 0, 1, 1
     0, 0, 1, 1
     1, 1, 0, 0
     1, 1, 0, 1
Representing these rows as binary numbers in base 10 the first three terms of the sequence are: 11, 3, 12, 13.
The next numbers in the sequence occur for m = 2, given a sequence of six numbers. The Jacobi 7 X 7 matrix has as its rows:
     1, 1, 0, 0, 0, 0, 0
     1, 1, 1, 0, 0, 0, 0
     0, 1, 1, 1, 0, 0, 0
     0, 0, 1, 1, 1, 0, 0
     0, 0, 0, 1, 1, 1, 0
     0, 0, 0, 0, 1, 1, 1
     0, 0, 0, 0, 0, 1, 1
Its inverse has as rows:
     1, 0, 1, 1, 0, 1, 1
     0, 0, 1, 1, 0, 1, 1
     1, 1, 0, 0, 0, 0, 0
     1, 1, 0, 1, 0, 1, 1
     0, 0, 0, 0, 0, 1, 1
     1, 1, 0, 1, 1, 0, 0
     1, 1, 0, 1, 1, 0, 1
These 7 latter rows from binary to base 10 give the next 7 terms of the sequence: 91, 27, 96, 107, 3, 108, 109.
Triangle T(n,k) begins:
    11,    3,   12,   13;
    91,   27,   96,  107,   3,  108,  109;
   731,  219,  768,  859,  27,  864,  875,  3,  876,  877;
  5851, 1755, 6144, 6875, 219, 6912, 7003, 27, 7008, 7019, 3, 7020, 7021;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Boston, 2nd Ed., 12th printing, 2002, pp. 24-25.
P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Boston, 1985, p. 35.
J. P. Melo, Reversibility of John von Neumann cellular automata, M.Sc. Thesis, Division of Computer Science, Instituto Tecnológico de Aeronáutica, 1997 (in Portuguese), p. 18.

Nei Y. Soma, Rows n = 1..30, flattened
K. Sutner, Linear Cellular Automata and the Garden-of-Eden, The Mathematical Intelligencer, 11(2), 1989, pp. 49-53; see p. 52

Column k=1 gives A245599(n+1).
Column k=2 gives A083713.
Column k=3 gives A204623.
T(n,3n-1) gives A010701.
Cf. A038184 One-dimensional cellular automaton (Rule 150) in a tape with 3m cells has as adjacency matrix the Jacobi matrices, 3m X 3m, with 1s on the lower, main and upper diagonals and the operations on it are in GF(2). And A363146 for the inverse of Jacobi matrices 3m X 3m, with 1s on the lower, main, and upper diagonals in GF(2).

T:= n-> (M-> seq(add(abs(M[j, n*3+1-i])*2^i, i=0..n*3), j=1..n*3+1))
               (Matrix(n*3+1, (i, j)-> `if`(abs(i-j)<2, 1, 0))^(-1)):
seq(T(n), n=1..6);  # Alois P. Heinz, May 20 2023

sequence = {};
For[k = 1, k <= 50, k++, {
  n = 3*k + 1;
  J = ConstantArray[0, {n, n}];
  For[i = 1, i < n, i++,
   J[[i, i]] = J[[i + 1, i]] = J[[i, i + 1]] = 1];
  J[[1, 1]] = J[[n, n]] = 1;
  InvJ = Mod[Inverse[J], 2];
  For[i = 1, i <= n, i++, AppendTo[sequence, FromDigits[InvJ[[i]], 2]]]
  }
 ]
sequence

row(n)=my(m=3*n+1, A=lift(matrix(m, m, i, j, Mod(abs(i-j)<=1, 2))^(-1))); vector(m, i, fromdigits(A[i,], 2)) \\ Andrew Howroyd, May 20 2023

The recurrence has as its base:
     r(1, 1) = 11;
     r(2, 1) =  3;
     r(3, 1) = 12;
     r(4, 1) = 13.
For 2 <= k <= m, and i = 1, 2, 3, ..., 3k - 2:
    r(i, k) = 8*r(i, k-1) + r(2, 1) (i != 0 (mod 3)).
And r(3k-1, k) = r(2, 1);
    r(3k,   k) = 8*r(3(k-1), k-1) + r(3,1);
    r(3k+1, k) = 8*r(3(k-1), k-1) + r(4,1).

cp's OEIS Frontend

A208258 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than one of its immediate leftward or upward neighbors.

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A270568 Expansion of g.f. (1+4x)/(1-8x).

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A363099 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).

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cp's OEIS Frontend

A208258 T(n,k)=Number of nXk 0..2 arrays with new values 0..2 introduced in row major order and no element equal to more than one of its immediate leftward or upward neighbors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A270568 Expansion of g.f. (1+4*x)/(1-8*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A363099 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A270568 Expansion of g.f. (1+4x)/(1-8x).