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A088858 Define a Fibonacci-type sequence to be one of the form s(0) = s_1 >= 1, s(1) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.

%I A088858 #28 Jan 12 2025 09:21:51
%S A088858 1,2,3,3,4,3,4,5,4,4,5,4,6,5,4,5,5,6,5,5,7,5,6,5,5,6,5,6,7,5,6,5,6,8,
%T A088858 5,6,7,6,6,5,6,7,6,6,7,6,8,6,6,7,6,6,7,6,9,6,6,7,6,8,7,6,7,6,6,7,6,8,
%U A088858 7,6,7,6,8,7,6,9,7,6,7,6,8,7,6,7,7,8,7,6,10,7
%N A088858 Define a Fibonacci-type sequence to be one of the form s(0) = s_1 >= 1, s(1) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.
%C A088858 A033192(k) is the number of integers m such that a(m) = k. - _Michel Marcus_, Aug 03 2017
%H A088858 Michel Marcus, <a href="/A088858/b088858.txt">Table of n, a(n) for n = 1..1000</a>
%H A088858 J. H. E. Cohn, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-1/cohn.pdf">Recurrent Sequences Including N</a>, Fib. Quart., 29-1, 1991.
%H A088858 T. Denes, <a href="https://doi.org/10.1016/S0012-365X(03)00207-3">Problem 413</a>, Discrete Math. 272 (2003), 302 (but there are several errors in the table given there).
%H A088858 James P. Jones and Péter Kiss, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AAPASM_25_from21to37.pdf">Representation of integers as terms of a linear recurrence with maximal index</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25. (1998) pp. 21-37.
%F A088858 For n>1, a(n) is the largest integer r>1 such that ceiling(((-1)^r*Fibonacci(r-2)*n + 1)/Fibonacci(r-1)) <= floor(((-1)^r*Fibonacci(r-1)*n - 1)/Fibonacci(r)). See Theorem 2.12 in Jones & Kiss. - _Michel Marcus_, Aug 02 2017
%t A088858 max = 12; s[n_] := (1/2)*((3*s1 - s2)*Fibonacci[n] + (s2 - s1)*LucasL[n]); a[n_] := Reap[Do[If[s[m] == n, Sow[m - 1]], {m, 1, max}, {s1, 1, max}, {s2, 1, max}]][[2, 1]] // Max; Table[a[n], {n, 1, 90}] (* _Jean-François Alcover_, Jan 15 2013 *)
%o A088858 (PARI) a(n) = {if (n==1, return (1)); r = 2; while (ceil(((-1)^r*fibonacci(r-2)*n + 1)/fibonacci(r-1)) <= floor(((-1)^r*fibonacci(r-1)*n - 1)/fibonacci(r)), r++); r-1;} \\ _Michel Marcus_, Aug 02 2017
%Y A088858 Cf. A033192, A088527 (which is a(n)+1).
%K A088858 nonn,easy
%O A088858 1,2
%A A088858 _Don Reble_, Nov 20 2003