%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