A102243 - OEIS

%I #185 Jul 22 2022 04:35:41

%S 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,

%T 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,

%U 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

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

%C George Bergman wrote his paper when he was 12. Mike Wallace interviewed him when Bergman was 14. - _Robert G. Wilson v_, Mar 14 2014

%H Robert G. Wilson v, <a href="/A102243/b102243.txt">Table of n, a(n) for n = 3..1002</a> (offset adapted by _Georg Fischer_, Jan 24 2019)

%H George Bergman, <a href="https://www.jstor.org/stable/3029218">A number system with an irrational base</a>, Math. Mag. 31 (1957), pp. 98-110.

%H Chittaranjan Pardeshi, <a href="/A102243/a102243.txt">100000 digits of Pi in golden base</a>

%H Chittaranjan Pardeshi, <a href="/A000796/a000796.pdf">BBP-Like formula for Pi in Golden Ratio Base Phi</a>

%H Mike Wallace, <a href="https://www.jstor.org/stable/3029389">Mike Wallace Asks George Bergman: What Makes a Genius Tick?</a>, Math. Mag. 31 (1958), p. 282.

%F 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

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

%t RealDigits[Pi, GoldenRatio, 111][[1]] (* _Robert G. Wilson v_, Feb 26 2010 *)

%o (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,",")))

%o (PARI)

%o alist(len) = {

%o     my(phi=quadgen(5), n=-1, pi=4/phi, gap=phi^3, hi=pi+gap, t=0, w=phi^3);

%o     vector(len, i,

%o         w = w/phi;

%o         while(t+w < hi && t+w > pi,

%o             n   = n + 1;

%o             pi += phi^(-12*n) * (

%o                     8 * phi^-3  / (12*n+3)

%o                   + 4 * phi^-5  / (12*n+5)

%o                   - 4 * phi^-7  / (12*n+7)

%o                   - 8 * phi^-9  / (12*n+9)

%o                   - 4 * phi^-11 / (12*n+11)

%o                   + 4 * phi^-13 / (12*n+13));

%o             gap /= phi^12;

%o             hi   = pi + gap);

%o         if( t+w <= pi, t += w; 1, 0))};

%o alist(1000) \\ _Chittaranjan Pardeshi_, May 18 2022

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

%K cons,base,nonn

%O 3

%A _Benoit Cloitre_, Feb 18 2005

%E Offset corrected by _Lee A. Newberg_, Apr 13 2018