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A342148
Infinite square matrix A(m,n) = F(m) mod n, m,n >= 1, where F = Fibonacci = A000045, read by falling antidiagonals.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
M. F. Hasler, Jul 12 2021
STATUS
approved