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%I #33 Aug 02 2023 13:57:48
%S 2,4,7,11,13,22,23,332,560,565,566,568,616,618,1161,1163,1167,1194,
%T 1298,1317,1321,1329,1360,1370,1371,1373,1374,1376,1386,1391,1503,
%U 1506,1081319,1081322,1081349,1081353,1081356,1081358,1081363,1081365,1081367,1081376,1081379
%N Define a sequence of fractions by x(m+1) = (x(m)^2+x(m)-1)/(x(m)+1) with x(0)=1; sequence gives values of k such that sgn(x(k))<>sgn(x(k-1)).
%H Hugo Pfoertner, <a href="/A064690/b064690.txt">Table of n, a(n) for n = 1..1191</a>
%H Sean A. Irvine <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a064/A064690.java">Java program</a> (github)
%H Hugo Pfoertner, <a href="/A064690/a064690.png">x values plotted against m</a>.
%e x(0)=1, x(1) = 1/2, x(2) = -1/6 so a(1) = 2 because sgn(x(1)) <> sgn(x(2)).
%p x := 1;
%p for n from 1 do
%p xold := x;
%p x := x-1/(x+1) ;
%p if x*xold < 0 then
%p print(n) ;
%p end if;
%p end do: # _R. J. Mathar_, Jul 23 2023
%o (Python)
%o from itertools import count, islice
%o def A064690_gen(): # generator of terms
%o ap, aq, sa = 1, 1, True
%o for n in count(1):
%o if (r:=ap+aq) > 0:
%o bq = aq*r
%o bp = ap*r-aq**2
%o else:
%o bq = -aq*r
%o bp = aq**2-ap*r
%o if (sa is not (sb:=bp>0)):
%o yield n
%o ap, aq, sa = bp, bq, sb
%o A064690_list = list(islice(A064690_gen(),7)) # _Chai Wah Wu_, Jul 24 2023
%o (PARI) A64690=List(); m64690=0; x64690=1.; A064690(n)={while(n>#A64690, until(sign(x64690)!=sign(x64690-=1/(x64690+1)), m64690++); listput(A64690, m64690)); A64690[n]} \\ Precision must be set to twice the number of desired terms: correct up to a(21) with standard precision \p38, up to a(56) with \p99, up to a(104) with \p199. - _M. F. Hasler_, Jul 24 2023
%K nonn
%O 1,1
%A Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 16 2001
%E a(13) onward corrected by _Sean A. Irvine_ and _Hugo Pfoertner_, Jul 23 2023
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