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%I #37 Dec 12 2013 14:01:20
%S 2,-6,34,1154,1331714,1773462177794,3145168096065837266706434,
%T 9892082352510403757550172975146702122837936996354
%N Optimal ascending continued fraction expansion of sqrt(2)-1.
%C See A228929 for the definition of "optimal ascending continued fraction".
%C Conjecture: The terms from a(3) are all positive and can be generated by the recurrence relation a(k+1) = a(k)^2 - 2.
%C This relation was studied by Lucas with reference to Engel expansion.
%C This recurrence is not peculiar of sqrt(2) but is present in the expansion of the square root of many other numbers, starting from some term onward, but not for all numbers. Here is a list of the numbers in range 1..200 having the recurrence: 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 45, 47, 48, 50, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 82, 83, 84, 87, 88, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 110, 111, 112, 114, 117, 119, 120, 122, 123, 124, 126, 128, 130, 132, 133, 135, 136, 138, 140, 141, 142, 143, 145, 146, 147, 148, 150, 152, 155, 156, 158, 162, 164, 165, 167, 168, 170, 171, 174, 175, 178, 180, 182, 183, 185, 187, 188, 189, 192, 194, 195, 197, 198, 200
%C Essentially the same as A003423. - _R. J. Mathar_, Sep 21 2013
%H P. Bala, <a href="https://oeis.org/A220335/a220335.pdf">A modified Engel expansion for certain quadratic irrationals</a>
%H Giovanni Artico, <a href="/A228931/a228931.pdf">Proof of the conjecture</a>
%F a(n) = a(n-1)^2 - 2, for n > 2.
%F For n>2, a(n) = (sqrt(2)+1)^(2^(n-1)) + (sqrt(2)-1)^(2^(n-1)). - _Vaclav Kotesovec_, Sep 20 2013
%e sqrt(2)=1+1/2*(1-1/6*(1+1/34*(1+1/1154*(1+1/1331714*(1+1/1773462177794*(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 sqrt(2)-1
%p ArticoExp(sqrt(2),8)
%t Flatten[{2, RecurrenceTable[{a[n] == a[n-1]^2 - 2, a[2] == -6}, a, {n, 2, 10}]}] (* _Vaclav Kotesovec_, Sep 20 2013 *)
%Y Cf. A228929, A220335.
%K sign,cofr,easy
%O 1,1
%A _Giovanni Artico_, Sep 09 2013
%E Added a pdf file with a proof of the conjecture by _Giovanni Artico_
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