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A185443 Engel expansion of A060997 = 1.433127...

%I A185443 #15 Feb 16 2025 08:33:13
%S A185443 1,3,4,6,6,10,10,10,21,66,207,722,6563,25007,372733,2028763,5472218,
%T A185443 41430101,75142985,192675195,201216921,925285050,935598827,2288358581,
%U A185443 2346034092,26271379744,41588896504,152594692251,529451874660
%N A185443 Engel expansion of A060997 = 1.433127...
%H A185443 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%p A185443 Digits := 5000:
%p A185443 a0 := evalf(BesselI(0,2)/BesselI(1,2)):
%p A185443 f1 := proc(n) local i, an, u, a:
%p A185443 an := [ ]:
%p A185443 u := n:
%p A185443 for i from 1 to 30 do
%p A185443    a := ceil(1/u):
%p A185443    an := [ op(an), a ]:
%p A185443    u := u * a - 1:
%p A185443 od:
%p A185443 RETURN (an): end:  f1(a0);
%o A185443 (PARI) CFB(v)={ \\ converts a continued fraction to a number
%o A185443 my(x=v[#v]*1.);
%o A185443 forstep(i=#v-1,1,-1,
%o A185443 x = v[i] + x^-1;
%o A185443 );
%o A185443 x
%o A185443 };
%o A185443 Engel(x)={
%o A185443 my(v=List(),t);
%o A185443 while(1,
%o A185443 trap(,
%o A185443 return(Vec(v))
%o A185443 ,
%o A185443 t = ceil(1/x)
%o A185443 );
%o A185443 listput(v,t);
%o A185443 x = (x * t) - 1
%o A185443 )
%o A185443 };
%o A185443 \p 500
%o A185443 Engel(CFB(vector(500,i,i)))
%Y A185443 Cf. A006784.
%K A185443 nonn
%O A185443 1,2
%A A185443 _Jani Melik_, Feb 04 2011
%E A185443 gp script from _Charles R Greathouse IV_, Feb 06 2011