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A330268
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Greedy base sqrt(5) expansion of 1/2.
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1
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1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 2
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OFFSET
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0,30
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COMMENTS
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To write a real number in a non-integer base, the greedy algorithm takes the largest possible integer digit in the range 0 <= digit < base at each digit position from high to low. Kempner considers this as a "canonical" representation and for base sqrt(5) gives the present sequence as an example canonical 1/2 = .100101... (section IV, page 616).
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LINKS
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FORMULA
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Sum_{n>=0} a(n)/sqrt(5)^(n+1) = 1/2.
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EXAMPLE
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0.100101101110100010101001000102000... = 1/sqrt(5) + 1/sqrt(5)^4 + 1/sqrt(5)^6 + ...
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PROG
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(PARI) a_vector(len) = my(v=vector(len), sqrt5=quadgen(20), r=1/2); for(i=1, len, r*=sqrt5; v[i]=floorQuad(r); r-=v[i]); v;
floorQuad(x) = my(ret=0); while(x>=1, ret++; x--); ret; \\ pending GP 2.13.x which will allow floor() of quads
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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