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A072030
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Triangle T(a,b) read by rows giving number of steps in simple Euclidean algorithm for GCD(a,b) (a > b >= 1).
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2
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1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 2, 1, 2, 5, 6, 4, 4, 4, 4, 6, 7, 3, 4, 1, 4, 3, 7, 8, 5, 2, 5, 5, 2, 5, 8, 9, 4, 5, 3, 1, 3, 5, 4, 9, 10, 6, 5, 5, 6, 6, 5, 5, 6, 10, 11, 5, 3, 2, 5, 1, 5, 2, 3, 5, 11, 12, 7, 6, 6, 5, 7, 7, 5, 6, 6, 7, 12, 13, 6, 6, 4, 6, 4, 1, 4, 6, 4, 6, 6, 13, 14, 8, 4, 6, 2, 3, 8
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| For example <11,3> -> <8,3> -> <5,3> -> <3,2> -> <2,1> -> <1,1> has length 5.
The length is extended to pairs <b,a> with b < a by T(b,a)=T(a,b). The length function can be defined inductively by: T(a,a)=0, T(a,b)=T(b,a), T(a+b,a)=T(a+b,b)=T(a,b)+1.
The simple Euclidean algorithm is the Euclidean algorithm without divisions. Given a pair <a,b> of positive integers with a>=b, let <a^(1),b^(1)> = <max(a-b,b),min(a-b,b)>. This is iterated until a^(m)=b^(m), which is then the greatest common divisor of a and b. T(a,b) gives length or number of steps m.
Note that row n has n-1 terms; the length of computing GCD(n,n), which is 0, is not in the triangle. - T. D. Noe, Oct 29 2007
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LINKS
| T. D. Noe, Rows n=2..50 of triangle, flattened
James C. Alexander, The Entropy of the Fibonacci Code (1989).
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EXAMPLE
| 1; 2, 2; 3, 1, 3; 4, 3, 3, 4; 5, 2, 1, 2, 5; 6, 4, 4, 4, 6; 7, 3, 4, 1, 4, 3, 7; ...
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PROG
| (PARI) {T(n, k) = if( n<1 | k<1, 0, if( n==k, 0, if( n<k, T(k, n), 1 + T(k, n-k))))}
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CROSSREFS
| Row sums are A072031. Cf. A003989, A050873.
Sequence in context: A143182 A128715 A113881 * A205456 A080045 A191384
Adjacent sequences: A072027 A072028 A072029 * A072031 A072032 A072033
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| Michael Somos, Jun 07 2002
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