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.
16 = 13 + 3, so row 16 is 13, 3. [Corrected by _Sean A. Irvine_, Oct 14 2020]
The first few rows are:
0;
1;
2;
3;
3, 1;
5;
5, 1;
5, 2;
8;
8, 1;
8, 2;
...
Row 1000000 is 832040,121393,46368,144,55. Indeed, the Maple program yields in no time Z(1000000) = {55,144,46368,121393,832040}. - _Emeric Deutsch_, Oct 22 2014
a035516 n k = a035516_tabf !! n !! k a035516_tabf = map a035516_row [0..] a035516_row 0 = [0] a035516_row n = z n $ reverse $ takeWhile (<= n) a000045_list where z 0 _ = [] z x (f:fs'@(_:fs)) = if f <= x then f : z (x - f) fs else z x fs' -- Reinhard Zumkeller, Mar 10 2013
with(combinat): Z := proc (n) local F, LF, A, m: F := proc (n) options operator, arrow: fibonacci(n) end proc: LF := proc (m) local i: for i from 0 while F(i) <= m do end do: F(i-1) end proc: A := {}: m := n: while 0 < m do A := `union`(A, {LF(m)}): m := m-LF(m) end do: A end proc: # The Maple program, with the command Z(n), yields the set of the Fibonacci numbers in the Zeckendorf representation of n (terms in {} are in reverse order). - Emeric Deutsch, Oct 21 2014
t = Fibonacci /@ Range@ 12; Table[If[MemberQ[t, n], {n}, Most@ MapAt[# + 1 &, Abs@ Differences@ FixedPointList[# - First@ Reverse@ TakeWhile[t, Function[k, # >= k]] &, n], -1]], {n, 41}] // Flatten (* faster, or *)
t = Fibonacci /@ Range@ 12; {{0}}~Join~Table[First@ Select[ Select[ IntegerPartitions@ n, Times @@ Boole@ Map[MemberQ[t, #] &, #] == 1 &], Times @@ Boole@ Map[# > 1 &, Abs@ Differences@ Map[Position[t, #][[1, 1]] &, #, {1}]] == 1 &], {n, 41}] // Flatten (* Michael De Vlieger, May 17 2016 *)
a(33) = 21*8*3*1 because 33 = 21+8+3+1.
a273156 = product . a035516_row
A273156 := proc(n) local nred,a,f ; if n = 0 then 0; else nred := n ; a := 1 ; while nred > 1 do f := A087172(nred) ; a := a*f ; nred := nred-f ; end do: a ; end if; end proc: # R. J. Mathar, May 17 2016
t = Fibonacci /@ Range@ 21; {0}~Join~Table[Times @@ If[MemberQ[t, n], {n}, Most@ MapAt[# + 1 &, Abs@ Differences@ FixedPointList[# - First@ Reverse@ TakeWhile[t, Function[k, # >= k]] &, n], -1]], {n, 58}] (* Michael De Vlieger, May 17 2016 *)
a[0]=0; a[n_]:=Block[{m=n, p=1, f, k=0}, While[Fibonacci@ ++k <= n]; While[ m>1, f= Fibonacci@ --k; If[ f<=m, m-=f; p*=f]]; p]; Array[a, 80, 0] (* Giovanni Resta, May 17 2016 *)
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