A228931 Optimal ascending continued fraction expansion of sqrt(2)-1.

2, -6, 34, 1154, 1331714, 1773462177794, 3145168096065837266706434, 9892082352510403757550172975146702122837936996354

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

Table of n, a(n) for n=1..8.

P. Bala, A modified Engel expansion for certain quadratic irrationals

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: A353536 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