A088527 -id:A088527 - cp's OEIS frontend

A306574 Incorrect version of A088527.

Original entry on oeis.org

2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 5, 7, 5, 5, 6, 6, 7, 6, 6, 8, 6, 6, 6

Offset: 1

dead

Claimed to be the maximum k such that n is the k-th term in some Fibonacci-type sequence.

Tamás Dénes, Problem 413. Fibonacci index of natural numbers, Research problems / Discrete Mathematics 272 (2003) 301-306 (but there are several errors in the table given there).

Incorrect version of A088527.

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.

Original entry on oeis.org

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, 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, 7, 6, 7, 6, 8, 7, 6, 9, 7, 6, 7, 6, 8, 7, 6, 7, 7, 8, 7, 6, 10, 7

Offset: 1

Don Reble, Nov 20 2003

A033192(k) is the number of integers m such that a(m) = k. - Michel Marcus, Aug 03 2017

Michel Marcus, Table of n, a(n) for n = 1..1000
J. H. E. Cohn, Recurrent Sequences Including N, Fib. Quart., 29-1, 1991.
T. Denes, Problem 413, Discrete Math. 272 (2003), 302 (but there are several errors in the table given there).
James P. Jones and Péter Kiss, Representation of integers as terms of a linear recurrence with maximal index, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25. (1998) pp. 21-37.

Cf. A033192, A088527 (which is a(n)+1).

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 *)

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

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

A325171 Down-integers: integers k such that w_(s+1) = floor(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.

Original entry on oeis.org

2, 5, 7, 9, 10, 12, 13, 15, 18, 23, 26, 28, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 99, 102, 104, 107, 112, 115, 120, 123, 125, 128, 133, 136, 138, 141, 146, 149

Offset: 1

Michel Marcus, Apr 04 2019

nonn

An n-slow Fibonacci walk is a Fibonacci-like sequence that needs a maximum number of steps, s (see A088527), to reach n, and w_(s+1) will be the next term of this sequence. See Chung et al. for further explanation.

Fan Chung, Ron Graham, and Sam Spiro, Slow Fibonacci Walks, arXiv:1903.08274 [math.NT], 2019. See pp. 3-4.

Cf. A001622 (phi), A088527, A325172.

nbs(i, j, n) = {my(nb = 2, ij); until (j >= n, ij = i+j; i = j; j = ij; nb++); if (j==n, nb, -oo);}
dofib(i, j, nb) = {if (nb==2, return (j)); for (k=3, nb, ij = i + j; i = j; j = ij;); return (j);}
s(n) = {my(nb = 2, k); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k> nb, nb = k););); nb;} \\ A088527
isdown(n) = {my(nb = s(n)); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k == nb, w = dofib(i, j, nb+1); if (w == floor(n*((1+sqrt(5))/2)), return (1));););); return (0);}

A325172 Up-integers: integers k such that w_(s+1) = ceiling(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.

Original entry on oeis.org

3, 4, 6, 8, 11, 14, 16, 17, 19, 20, 21, 22, 24, 25, 27, 29, 30, 32, 35, 37, 40, 42, 45, 50, 53, 55, 58, 61, 63, 66, 69, 71, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 109, 110, 111, 113, 114, 116, 117, 118, 119, 121, 122, 124

Offset: 1

Michel Marcus, Apr 04 2019

nonn

An n-slow Fibonacci walk is a Fibonacci-like sequence that needs a maximum number of steps, s (see A088527), to reach n, and w_(s+1) will be the next term of this sequence. See Chung et al. for further explanation.

Fan Chung, Ron Graham, and Sam Spiro, Slow Fibonacci Walks, arXiv:1903.08274 [math.NT], 2019. See pp. 3-4.

Cf. A001622 (phi), A088527, A325171.

nbs(i, j, n) = {my(nb = 2, ij); until (j >= n, ij = i+j; i = j; j = ij; nb++); if (j==n, nb, -oo);}
dofib(i, j, nb) = {if (nb==2, return (j)); for (k=3, nb, ij = i + j; i = j; j = ij;); return (j);}
s(n) = {my(nb = 2, k); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k> nb, nb = k););); nb;} \\ A088527
isup(n) = {my(nb = s(n)); for (i=1, n, for (j=1, n, k = nbs(i, j, n); if (k == nb, w = dofib(i, j, nb+1); if (w == ceil(n*((1+sqrt(5))/2)), return (1));););); return (0);}

cp's OEIS Frontend

A306574 Incorrect version of A088527.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A325171 Down-integers: integers k such that w_(s+1) = floor(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A325172 Up-integers: integers k such that w_(s+1) = ceiling(phi*k) for some k-slow Fibonacci walk, with phi=(1+sqrt(5))/2. See comments for further explanation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI