This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A228931 #37 Dec 12 2013 14:01:20
%S A228931 2,-6,34,1154,1331714,1773462177794,3145168096065837266706434,
%T A228931 9892082352510403757550172975146702122837936996354
%N A228931 Optimal ascending continued fraction expansion of sqrt(2)-1.
%C A228931 See A228929 for the definition of "optimal ascending continued fraction".
%C A228931 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 A228931 This relation was studied by Lucas with reference to Engel expansion.
%C A228931 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 A228931 Essentially the same as A003423. - _R. J. Mathar_, Sep 21 2013
%H A228931 P. Bala, <a href="https://oeis.org/A220335/a220335.pdf">A modified Engel expansion for certain quadratic irrationals</a>
%H A228931 Giovanni Artico, <a href="/A228931/a228931.pdf">Proof of the conjecture</a>
%F A228931 a(n) = a(n-1)^2 - 2, for n > 2.
%F A228931 For n>2, a(n) = (sqrt(2)+1)^(2^(n-1)) + (sqrt(2)-1)^(2^(n-1)). - _Vaclav Kotesovec_, Sep 20 2013
%e A228931 sqrt(2)=1+1/2*(1-1/6*(1+1/34*(1+1/1154*(1+1/1331714*(1+1/1773462177794*(1+.....))))))
%p A228931 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 A228931 # List the first 8 terms of the expansion of sqrt(2)-1
%p A228931 ArticoExp(sqrt(2),8)
%t A228931 Flatten[{2, RecurrenceTable[{a[n] == a[n-1]^2 - 2, a[2] == -6}, a, {n, 2, 10}]}] (* _Vaclav Kotesovec_, Sep 20 2013 *)
%Y A228931 Cf. A228929, A220335.
%K A228931 sign,cofr,easy
%O A228931 1,1
%A A228931 _Giovanni Artico_, Sep 09 2013
%E A228931 Added a pdf file with a proof of the conjecture by _Giovanni Artico_