Jörg> I'll edit the sequence. If you do not see the edit within 10 days, remind me. Note the L-system I gave for the binary expansion, giving a divisionless method for computation. Best, jj I think it's been 10 days. There's also this glitch: "%t Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 6] (* _Robert G. Wilson v_, Mar 09 2005 *)" always gets the last bit wrong: In[207]:= Nest[Flatten[# /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 1] Out[207]= {1, 1, 1} In[208]:= Nest[Flatten[# /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 2] Out[208]= {1, 1, 0, 1, 1, 0, 1, 1, 0} And Nest[Join[#, #, MapAt[1 - # &, #, -1]] &, {1}, 4] is shorter. --rwg On Tue, Apr 15, 2014 at 3:13 PM, Bill Gosper <billgosper@gmail.com> wrote: (A014578 <http://oeis.org/A014578>) Out[665]= k (-1) Sum[--------, {k, 0, ∞}] k 3 2 - 1 In[666]:= RealDigits[% /. ∞-> 4, 8] Out[666]= {{{6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 7}}, 0} Note that this and the other partial sums are actual exemplars of Roth's criterion. In[685]:= FromDigits[%666, 8] Out[685]= 2077175852086666748456887/2417851639229258349412351 In[686]:= $MaxExtraPrecision = 9999; N[Round[1/(% - %665)], 239] N::meprec: Internal precision limit $MaxExtraPrecision = 9999.` reached while evaluating Round[1/((2077175852086666748456887/2417851639229258349412351)-\!\(\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)]\*FractionBox[SuperscriptBox[\((\(-1\))\), \(k\)], \(\(-1\) + Power[<<1>>]\)]\))]. >> (I don't understand this warning. 239 and 9999 are ridiculous overkill. Varying them (above a safe minimum) has no effect on the answer or the warning.) Out[686]= 14134776518227074636666380005943348126619871175004951664972849610340958207 In[697]:= Denominator[%685]^3 Out[697]= \ 14134776518227074636666362467923700155784852730794212624494301514198679551 where all that is needed is Denominator^(2+ϵ) (infinitely often). (As computed by the Life pattern x = 50, y = 50, rule = B3/S23 [...]) Tighter: #CXRLE #C Thue (really Roth) generator #C #C Four puffers produce gliders, whose collisions produce four #C lines of blinkers, each representing the transcendental number #C .110110111110110111110110110..., where the n'th bit is the #C the mod 2 of 1 + the largest power of 3 dividing n. #C Built by Bill Gosper, in or before 1987 x = 50, y = 50, rule = B3/S23 26b3o11b3o$26bo2bo10bo2bo$26bo6b3o4bo$26bo5bo2bo4bo$27bo7bo5bo2$bo8b2o $o8b4o$o3bo3b2ob2o28bo$4o5b2o29b3o$40bob2o$41b3o$41b2o2$2b3o$2bo$2bo$ 3bo$24b3o$24bo2bo$bo22bo$o23bo$o3bo14bo5bo$4o14bo$18bo3bo5b4o$18b4o5bo 3bo$31bo14b4o$24bo5bo14bo3bo$25bo23bo$25bo22bo$22bo2bo$23b3o$46bo$47bo $47bo$45b3o2$7b2o$6b3o$6b2obo$7b3o29b2o5b4o$8bo28b2ob2o3bo3bo$37b4o8bo $38b2o8bo2$8bo5bo7bo$9bo4bo2bo5bo$9bo4b3o6bo$6bo2bo10bo2bo$7b3o11b3o! --rwg