A054770 (Conway's version of the Kimberling-table)

[Wouter Meeussen]

decompose an integer into non-adjacent fibs, use fib-indices

fibodec[50]
{9, 7, 4}

fib(9)+fib(7)+fib(4) = 34+13+3 = 50

fibodec[10^64]

{307, 305, 303, 299, 291, 289, 285, 283, 279, 277, 275, 270, 264, 262, 257, 
253, 250, 248, 246, 242, 240, 238, 235, 233, 228, 224,  221, 218, 216, 212, 
208, 201, 197, 185, 183, 181, 179, 177, 175, 171, 164, 159, 157, 155, 152, 
146, 140, 137, 133, 125, 118, 114,  110, 108, 102, 98, 95, 84, 80, 78, 76, 
72, 69, 67, 65, 63, 60, 58, 53, 51, 43, 41, 38, 34, 32, 30, 27, 25, 22, 11, 
9, 7}

Plus @@ Fibonacci /@ %

10000000000000000000000000000000000000000000000000000000000000000

(checks out ok)

Now, subtract 1 from all fib-indices and you get the left neighbour in 
Conway's version of the Kimberling-table.
The leading column (JHC's "-1"-th column) is obtained by subtracting the 
last fib-index, resulting in a trailing zero.
The "para-fib" is just to its right, so you subtract the (last fibindex-1).
Put Humpty-Dumpty together again by Fibonacci( fib-index) and summing.


Last[fibodec[50]] - 1
((Plus @@ Fibonacci /@ #1 & ) /@ {#1 - Last[#1], #1 - Last[#1] + 1, #1} & 
)[fibodec[50]]

3                      (*   (last element -1) of fibodec[50]= {9, 7, 4} 
   *)

{7, 12, 50}          (*   7-th row, para-fib[7]=12,  so it's column 3 
    *)


Let's go a bit bigger :

Last[fibodec[10^64]] - 1
((Plus @@ Fibonacci /@ #1 & ) /@ {#1 - Last[#1], #1 - Last[#1] + 1, #1} & 
)[fibodec[10^64]]

6
{344418537486330266596288467532955303640193374749172077608320951, 
557280900008412143633053250748950582375265615538971029164110184,
  10000000000000000000000000000000000000000000000000000000000000000}

so it's row 344418...0951  , and that one's par-fib is 55728...0184, and 
10^64 is found in column 6.

not so hard, and beats doing it by hand.

Wouter.


------------------------------------------------------------------------  
---------------------------------------------------
fibodec[n_Integer] := Module[{it},
   it = FixedPointList[#1 - (Function[z, First[Select[Fibonacci[{1, 0, -1} 
+ Floor[2 + Log[GoldenRatio, z]]], #1 <= z & , 1]]])[
         #1] & , n, SameTest -> (#2 <= 0 & )]; (Floor[2 + Log[GoldenRatio, 
#1]] & ) /@ (Drop[it, -1] - Rest[it])]




Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
w.meeussen at vandemoortele.be