A102243 Expansion of Pi in golden base (i.e., in irrational base phi = (1+sqrt(5))/2) = A001622.

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0

Offset

3

Comments

George Bergman wrote his paper when he was 12. Mike Wallace interviewed him when Bergman was 14. - Robert G. Wilson v, Mar 14 2014

Links

Robert G. Wilson v, Table of n, a(n) for n = 3..1002 (offset adapted by Georg Fischer, Jan 24 2019)

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.

Chittaranjan Pardeshi, 100000 digits of Pi in golden base

Chittaranjan Pardeshi, BBP-Like formula for Pi in Golden Ratio Base Phi

Mike Wallace, Mike Wallace Asks George Bergman: What Makes a Genius Tick?, Math. Mag. 31 (1958), p. 282.

Formula

Pi = 4/phi + Sum_{n>=0} (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, May 16 2022

Example

Pi = phi^2 + 1/phi^2 + 1/phi^5 + 1/phi^7 + ... thus Pi = 100.0100101010010001010101000001010... in golden base.

Mathematica

RealDigits[Pi, GoldenRatio, 111][[1]] (* Robert G. Wilson v, Feb 26 2010 *)

Prog

(PARI) f=(1+sqrt(5))/2; z=Pi; b=0; m=100; for(n=1, m, c=ceil(log(z)/log(1/f)); z=z-1/f^c; b=b+1./10^c; if(n==m, print1(b, ", ")))
(PARI)
alist(len) = {
    my(phi=quadgen(5), n=-1, pi=4/phi, gap=phi^3, hi=pi+gap, t=0, w=phi^3);
    vector(len, i,
        w = w/phi;
        while(t+w < hi && t+w > pi,
            n   = n + 1;
            pi += phi^(-12*n) * (
                    8 * phi^-3  / (12*n+3)
                  + 4 * phi^-5  / (12*n+5)
                  - 4 * phi^-7  / (12*n+7)
                  - 8 * phi^-9  / (12*n+9)
                  - 4 * phi^-11 / (12*n+11)
                  + 4 * phi^-13 / (12*n+13));
            gap /= phi^12;
            hi   = pi + gap);
        if( t+w <= pi, t += w; 1, 0))};
alist(1000) \\ Chittaranjan Pardeshi, May 18 2022

Crossrefs

Cf. A000796, A001622, A004601, A004602, A004603, A004604, A004605, A004606, A004608, A006941, A062964, A068436, A068437, A068438, A068439, A068440, A238897.

Sequence in context: A181923 A339825 A290098 * A173859 A202108 A287530

Adjacent sequences: A102240 A102241 A102242 * A102244 A102245 A102246

Keyword

cons,base,nonn

Author

Benoit Cloitre, Feb 18 2005

Extensions

Offset corrected by Lee A. Newberg, Apr 13 2018

Status

approved