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A019586
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Vertical para-Fibonacci sequence: takes value i on later (i.e., b_j, j >= 2) terms of i-th Fibonacci sequence defined by b_0 = i, b_1 = [ tau(i+1) ].
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25
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0, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 6, 1, 7, 4, 0, 8, 5, 3, 9, 2, 10, 6, 1, 11, 7, 4, 12, 0, 13, 8, 5, 14, 3, 15, 9, 2, 16, 10, 6, 17, 1, 18, 11, 7, 19, 4, 20, 12, 0, 21, 13, 8, 22, 5, 23, 14, 3, 24, 15, 9, 25, 2, 26, 16, 10, 27, 6, 28, 17, 1, 29, 18, 11, 30, 7, 31, 19, 4, 32, 20, 12
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OFFSET
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1,6
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COMMENTS
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Gives number of row in Wythoff array that contains n. - Casey Mongoven, Sep 10 2005
For a method of generating this sequence that does not refer to the Wythoff array or Fibonacci numbers, see A003603. - Clark Kimberling, Oct 29 2009
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LINKS
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Table of n, a(n) for n=1..87.
J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences
Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
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FORMULA
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Says which row of Wythoff array (starting row count at 0) contains n.
If delete first occurrence of 0, 1, 2, 3, ... the sequence is unchanged.
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MAPLE
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A019586 := proc(n::posint)
local r, c, W ;
for r from 1 do
for c from 1 do
W := A035513(r, c) ;
if W = n then
return r-1 ;
elif W > n then
break ;
end if;
end do:
end do:
end proc:
seq(A019586(n), n=1..100) ; # R. J. Mathar, Aug 13 2021
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MATHEMATICA
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row[1] = row[2] = {1}; row[n_] := row[n] = Module[{ro, pos, lp, ins}, ro = row[n - 1]; pos = Position[ro, Alternatives @@ Intersection[ro, row[n - 2]]] // Flatten; lp = Length[pos]; ins = Range[lp] + Max[ro]; Do[ro = Insert[ro, ins[[i]], pos[[i]] + i], {i, 1, lp}]; ro];
Flatten[Array[row, 9] - 1] (* Jean-François Alcover, Jul 12 2016, after Clark Kimberling *)
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CROSSREFS
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Equals A003603(n) - 1.
Cf. Wythoff array: A035513.
Sequence in context: A025648 A025655 A022336 * A257962 A176095 A295508
Adjacent sequences: A019583 A019584 A019585 * A019587 A019588 A019589
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KEYWORD
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nonn,nice,easy,eigen
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AUTHOR
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N. J. A. Sloane and J. H. Conway
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EXTENSIONS
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Casey Mongoven reports that where the sequence reads 15,9,2,16,10,6,29,1,30,11,7,19,..., the 29 and 30 should be 17 and 18.
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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STATUS
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approved
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