A330268 Greedy base sqrt(5) expansion of 1/2.

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

Offset

0,30

Comments

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).

Links

Kevin Ryde, Table of n, a(n) for n = 0..9999

Aubrey J. Kempner, Anormal Systems of Numeration, American Mathematical Monthly, volume 43, number 10, December 1936, pages 610-617.

Formula

Sum_{n>=0} a(n)/sqrt(5)^(n+1) = 1/2.

Example

0.100101101110100010101001000102000... = 1/sqrt(5) + 1/sqrt(5)^4 + 1/sqrt(5)^6 + ...

Prog

(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

Crossrefs

Sequence in context: A341025 A364044 A269247 * A089310 A129753 A356324

Adjacent sequences: A330265 A330266 A330267 * A330269 A330270 A330271

Keyword

base,nonn

Author

Kevin Ryde, Dec 07 2019

Status

approved