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A342148
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Infinite square matrix A(m,n) = F(m) mod n, m,n >= 1, where F = Fibonacci = A000045, read by falling antidiagonals.
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2
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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
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OFFSET
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1,13
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COMMENTS
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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.
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LINKS
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FORMULA
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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.)
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EXAMPLE
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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 ...]
(...)
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PROG
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(PARI) A342148(m, n)=fibonacci(m)%n
row(n)=[A342148(m, n-m+1) | m<-[1..n]] \\ The n-th falling antidiagonal.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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