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 A228931 Optimal ascending continued fraction expansion of sqrt(2)-1. 5
 2, -6, 34, 1154, 1331714, 1773462177794, 3145168096065837266706434, 9892082352510403757550172975146702122837936996354 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS See A228929 for the definition of "optimal ascending continued fraction". Conjecture: The terms from a(3) are all positive and can be generated by the recurrence relation a(k+1) = a(k)^2 - 2. This relation was studied by Lucas with reference to Engel expansion. 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 Essentially the same as A003423. - R. J. Mathar, Sep 21 2013 LINKS Giovanni Artico, Proof of the conjecture FORMULA a(n) = a(n-1)^2 - 2, for n > 2. For n>2, a(n) = (sqrt(2)+1)^(2^(n-1)) + (sqrt(2)-1)^(2^(n-1)). - Vaclav Kotesovec, Sep 20 2013 EXAMPLE sqrt(2)=1+1/2*(1-1/6*(1+1/34*(1+1/1154*(1+1/1331714*(1+1/1773462177794*(1+.....)))))) MAPLE 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 # List the first 8 terms of the expansion of sqrt(2)-1 ArticoExp(sqrt(2), 8) MATHEMATICA Flatten[{2, RecurrenceTable[{a[n] == a[n-1]^2 - 2, a[2] == -6}, a, {n, 2, 10}]}] (* Vaclav Kotesovec, Sep 20 2013 *) CROSSREFS Cf. A228929, A220335. Sequence in context: A075272 A224913 A327038 * A101262 A135965 A018983 Adjacent sequences:  A228928 A228929 A228930 * A228932 A228933 A228934 KEYWORD sign,cofr,easy AUTHOR Giovanni Artico, Sep 09 2013 EXTENSIONS Added a pdf file with a proof of the conjecture by Giovanni Artico STATUS approved

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Last modified December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)