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A363146 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n X 3n 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).

%I A363146 #36 May 21 2023 19:28:55
%S A363146 3,7,6,27,59,48,3,55,54,219,475,384,27,443,432,3,439,438,1755,3803,
%T A363146 3072,219,3547,3456,27,3515,3504,3,3511,3510,14043,30427,24576,1755,
%U A363146 28379,27648,219,28123,28032,27,28091,28080,3,28087,28086,112347,243419,196608,14043,227035,221184,1755,224987,224256,219,224731
%N A363146 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n X 3n 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).
%C A363146 Each term of 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 have ascending and consecutive multiples of 3 sizes. 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 x 3m Jacobi matrices are 1 in GF(2), as proven by Sutner (1989), and alternate between -1 and 1 in R if m is odd or even, as proven by Melo (1987).
%C A363146 The recurrence, in line 3, uses the Iverson notation as presented in Graham, Knuth, and Patashnik (2002).
%C A363146 The proof of the correctness of that sequence of inverses is done by induction.
%D A363146 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Boston, 2nd Ed., 12th printing, 2002, pp. 24-25.
%D A363146 P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Boston, 1985, p. 35.
%D A363146 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.
%D A363146 K. Sutner, Linear Cellular Automata and the Garden-of-Eden, The Mathematical Intelligencer, 11(2), 1989, 49-53, p. 52.
%H A363146 Nei Y. Soma, <a href="/A363146/b363146.txt">Rows n = 1..30, flattened</a>
%F A363146 The recurrence has as its base: r(1, 1) = 3; r(2, 1) = 7 and r(3, 1) = 6;
%F A363146 For 2 <= k <= m, and i = 1, 2, ..., 3(k-1):
%F A363146     r(i, k) = 8*r(i, k-1) + r(1,1) (i != 0 (mod 3)).
%F A363146 And r(3k-2, k) = r(1, 1);
%F A363146     r(3k-1, k) = 8*r(3(k-1), k-1) + r(2,1);
%F A363146     r(3k,   k) = 8*r(3(k-1), (k-1)) + r(3, 1).
%e A363146 For n = 1, the Jacobi 3 X 3 matrix has as rows
%e A363146      1, 1, 0
%e A363146      1, 1, 1
%e A363146      0, 1, 1.
%e A363146 Its inverse has the rows
%e A363146      0, 1, 1
%e A363146      1, 1, 1
%e A363146      1, 1, 0.
%e A363146 Representing these rows as decimal numbers the first three terms of the sequence are: 3, 7, and 6.
%e A363146 The next terms in the sequence occur for n = 2, given a sequence of six numbers. The Jacobi 6 X 6 matrix has as its rows:
%e A363146       1, 1, 0, 0, 0, 0
%e A363146       1, 1, 1, 0, 0, 0
%e A363146       0, 1, 1, 1, 0, 0
%e A363146       0, 0, 1, 1, 1, 0
%e A363146       0, 0, 0, 1, 1, 1
%e A363146       0, 0, 0, 0, 1, 1.
%e A363146 Its inverse has as rows:
%e A363146       0, 1, 1, 0, 1, 1
%e A363146       1, 1, 1, 0, 1, 1
%e A363146       1, 1, 0, 0, 0, 0
%e A363146       0, 0, 0, 0, 1, 1
%e A363146       1, 1, 0, 1, 1, 1
%e A363146       1, 1, 0, 1, 1, 0.
%e A363146 These 6 latter rows from binary to decimal give the next 6 terms of the sequence: 27, 49, 48, 3, 55, and 54.
%e A363146 Triangle T(n,k) begins:
%e A363146      3,    7,    6;
%e A363146     27,   59,   48,   3,   55,   54;
%e A363146    219,  475,  384,  27,  443,  432,  3,  439,  438;
%e A363146   1755, 3803, 3072, 219, 3547, 3456, 27, 3515, 3504, 3, 3511, 3510;
%e A363146   ...
%p A363146 T:= n-> (M-> seq(add(abs(M[j, n*3-i])*2^i, i=0..n*3-1), j=1..n*3))
%p A363146                (Matrix(n*3, (i, j)-> `if`(abs(i-j)<2, 1, 0))^(-1)):
%p A363146 seq(T(n), n=1..10);  # _Alois P. Heinz_, May 20 2023
%t A363146 sequence = {};
%t A363146 m = 6;
%t A363146 For[k = 1, k <= m, k++, {
%t A363146   n = 3*k;
%t A363146   J = ConstantArray[0, {n, n}];
%t A363146   For[i = 1, i < n, i++,
%t A363146    J[[i, i]] = J[[i + 1, i]] = J[[i, i + 1]] = 1];
%t A363146   J[[1, 1]] = J[[n, n]] = 1;
%t A363146   InvJ = Mod[Inverse[J], 2];
%t A363146   For[i = 1, i <= n, i++, AppendTo[sequence, FromDigits[InvJ[[i]], 2]]]
%t A363146   }
%t A363146  ]
%t A363146 sequence
%o A363146 (PARI) row(n)=my(m=3*n, 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
%Y A363146 Column k=1 gives A083713.
%Y A363146 Column k=3 gives A083233.
%Y A363146 T(n,3n) gives A125837(n+1).
%Y A363146 T(n,3n-1) gives A083068.
%Y A363146 T(n,3n-2) gives A010701.
%Y A363146 Cf. A038184 one-dimensional cellular automaton (Rule 150) in a tape with 3n cells has as adjacency matrix the Jacobi matrices, 3n X 3n, with 1s on the lower, main and upper diagonals and the operations on it are in GF(2).
%K A363146 nonn,tabf
%O A363146 1,1
%A A363146 _Nei Y. Soma_, May 19 2023