A342148 Infinite square matrix A(m,n) = F(m) mod n, m,n >= 1, where F = Fibonacci = A000045, read by falling antidiagonals.

0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 3, 2, 0, 0, 1, 1, 2, 3, 1, 2, 1, 0, 1, 1, 2, 3, 0, 0, 1, 1, 0, 1, 1, 2, 3, 5, 3, 1, 0, 0, 0, 1, 1, 2, 3, 5, 2, 3, 1, 1, 1, 0, 1, 1, 2, 3, 5, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 5, 0, 6, 3, 4, 3, 2, 0, 0, 1, 1, 2, 3, 5, 8, 5, 0, 4, 0, 1, 0, 1, 0, 1, 1, 2, 3, 5, 8, 4, 5, 6, 1, 4, 0, 2, 1, 0

Offset

1,13

Comments

The determinant of the (upper left) n X n submatrix with row and column indices 2..n+1 is zero iff n >= 35. (Observation by Bill Gosper, math-fun mailing list.)

It immediately follows from the definition that the sequence of columns converges to the Fibonacci sequence A000045, and thus, each row is an eventually constant sequence. See Formula section for precise relations.

Links

Table of n, a(n) for n=1..120.

Bill Gosper, mysteriously vanishing sequence, math-fun mailing list, Jul 10 2021

Formula

A(m,n) = A000045(m) mod n; A(m,n) = A000045(m) for n > A000045(m);

A(m,n) = 0 for n = 1 or n = A000045(m);

for m < 3, A(m,n) = A057427(n+1) = [n > 1] = 1 iff n > 1, = 0 for n = 1. (Here [.] is the Iverson bracket.)

Example

The matrix reads:
   [0  1  1  1  1  1  1  1 ...]
   [0  1  1  1  1  1  1  1 ...]
   [0  0  2  2  2  2  2  2 ...]
   [0  1  0  3  3  3  3  3 ...]
   [0  1  2  1  0  5  5  5 ...]
   [0  0  2  0  3  2  1  0 ...]
   [0  1  1  1  3  1  6  5 ...]
   [0  1  0  1  1  3  0  5 ...]
   (...)

Prog

(PARI) A342148(m, n)=fibonacci(m)%n
row(n)=[A342148(m, n-m+1) | m<-[1..n]] \\ The n-th falling antidiagonal.

Crossrefs

Cf. A000045, A342149 (matrix without the first row and column).

Sequence in context: A316101 A211452 A035188 * A066295 A132004 A035154

Adjacent sequences: A342145 A342146 A342147 * A342149 A342150 A342151

Keyword

nonn,tabl

Author

M. F. Hasler, Jul 12 2021

Status

approved