A228933 - OEIS

%I #33 Apr 07 2023 04:28:32

%S 2,4,-18,322,103682,10749957122,115561578124838522882,

%T 13354478338703157414450712387359637585922,

%U 178342091698891843163466683840822101223162205277179656650156983624835803932590082

%N Optimal ascending continued fraction expansion of phi-1=1/phi=(sqrt(5)-1)/2 .

%C See A228929 for the definition of "optimal ascending continued fraction".

%C Conjecture: The golden ratio (phi) expansion exhibits from the fourth term the recurrence relation a(n) = a(n-1)^2 - 2 described in A228931.

%F a(n) = a(n-1)^2 - 2 for n>3.

%F For n>3, a(n) = (sqrt(5)+2)^(2^(n-2)) + (sqrt(5)-2)^(2^(n-2)). - _Vaclav Kotesovec_, Sep 20 2013

%F a(n) = 2*A081459(n-1) for n>3. - _Amiram Eldar_, Apr 07 2023

%e phi = 1+1/2*(1+1/4*(1-1/18*(1+1/322*(1+1/103682*(1+1/10749957122*(1+...))))))

%p ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc

%p # List the first 8 terms of the expansion of 1/phi

%p ArticoExp((sqrt(5)-1)/2,8)

%t Flatten[{2, 4, RecurrenceTable[{a[n] == a[n-1]^2 - 2, a[3] == -18}, a, {n, 3, 10}]}] (* _Vaclav Kotesovec_, Sep 20 2013 *)

%Y Cf. A001622, A081459, A094214, A228929, A228931, A228932.

%K sign,cofr

%O 1,1

%A _Giovanni Artico_, Sep 10 2013