Corrigendum: Reverse Newman: 2*ceiling(1/x) -1/x - 1 , E.g. In[272]:= NestWhileList[2*Ceiling[1/#] - 1/# - 1 &, 3/5, # > 0 &] Out[272]= {3/5, 4/3, 1/4, 3, 2/3, 3/2, 1/3, 2, 1/2, 1, 0} Omitendum: fusc: A002487 <http://oeis.org/A002487> Apologies. --rwg

http://oeis.org/A002487 mentions

It appears that the terms of this sequence are the number of odd entries in the diagonals of Pascal's triangle at 45 degrees slope. - Javier Torres (adaycalledzero(AT)hotmail.com), Aug 06 2009

A consequence is that the parity of Fusc(n) is the same as Fib(n), which has period 3: odd, odd, even, ... This suggests looking at other moduli, such as Fusc(n) mod 3, and also looking at other bits of Fusc(n) beyond the low order bit. Rich ------- Quoting Bill Gosper <billgosper@gmail.com>:

https://cp4space.wordpress.com/2013/10/24/enumerating-the-rationals/ means that the familiar fractal pattern of Pascal's Triangle mod 2 gosper.org/fusc.png enumerates the rationals! I.e., the column sums of the latter two figures give Dÿkstra's fusc: In[140]:= Table[Sum[Binomial[n - r, r], {r, 0, n}], {n, 0, 17}]

Out[140]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584}

In[139]:= Table[Sum[Mod[Binomial[n - r, r], 2], {r, 0, n}], {n, 0, 17}]

Out[139]= {1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4}

In[137]:= Ratios[%]

Out[137]= {1, 2, 1/2, 3, 2/3, 3/2, 1/3, 4, 3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 5, 4/5}

Among all the remarkable properties of Pascal's Triangle, I rank this near the top.

Adam's page mentions Newnham. Would that be the famous Alfred E, or is it simply Adam's Anglocentric spelling corrector?

Forgetting I had NeilB's Calkin-Wilf accelerator, I foolishly tried to count down 325/36 by reverse Newman: x -> 2 ceiling(x) - x - 1. Hours later, I thought to

In[118]:= ContinuedFraction[325/36]

Out[118]= {9, 36}

In[130]:= fromcf[{9, k}]

Out[130]= (1 + 9 k)/k

In[131]:= Table[%, {k, 11}]

Out[131]= {10, 19/2, 28/3, 37/4, 46/5, 55/6, 64/7, 73/8, 82/9, 91/10, 100/11}

In[136]:= noomct /@ %131

Out[136]= {1023, 1535, 2559, 4607, 8703, 16895, 33279, 66047, 131583, 262655, 524799}

In[133]:= FindSequenceFunction[%, k]

Out[133]= 511 + 2^(8 + k)

In[134]:= 9 + 1/36

Out[134]= 325/36

In[135]:= %% /. k -> 36

Out[135]= 17592186044927

promising a rather lengthy wait.

It is amazing how relatively congenial Newman is to convergents. Even for sqrt 7, which has period 4: In[259]:= Convergents[Sqrt[7], 16]

Out[259]= {2, 3, 5/2, 8/3, 37/14, 45/17, 82/31, 127/48, 590/223, 717/271, 1307/494, 2024/765, 9403/3554, 11427/4319, 20830/7873, 32257/12192}

In[260]:= tim[noomct[#]] & /@ %

During evaluation of In[260]:= 0.000038,0 . . .

During evaluation of In[260]:= 24.287843,0

During evaluation of In[260]:= 60.010885,0

Out[260]= {3, 7, 11, 27, 491, 1003, 1515, 3563, 62955, 128491, 194027, 456171, 8058347, 16446955, 24835563, 58389995}

It can do 32257/12192, but not 325/36. FindSequenceFunction readily but messily interlaces the four sequences:

In[261]:= FullSimplify[FindSequenceFunction[%, n], n \[Element] Integers]

Out[261]= (1/4064)(-3424 + 2^(7 n/4) (221 + 976 2^(1/4) + 498 Sqrt[2] + 188 2^(3/4) + (-1)^ n (221 - 976 2^(1/4) + 498 Sqrt[2] - 188 2^(3/4)) + (442 - 996 Sqrt[2]) Cos[(n \[Pi])/2] + 8 2^(1/4) (244 - 47 Sqrt[2]) Sin[(n \[Pi])/2]))

In[262]:= Table[%, {n, 16, 18}] // Simplify

Out[262]= {58389995, 1031468523, 2105210347}

In[263]:= Convergents[Sqrt[7], 18][[-3 ;; -1]]

Out[263]= {32257/12192, 149858/56641, 182115/68833}

In[264]:= tim[noomct[#]] & /@ Rest[%]

During evaluation of In[264]:= 1050.500020,0

During evaluation of In[264]:= 2107.729563,0

Out[264]= {1031468523, 2105210347} as predicted. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun