A256654 -id:A256654 - cp's OEIS frontend

A256655 R(k), the minimal alternating Fibonacci representation of k, concatenated for k = 0, 1, 2,....

0, 1, 2, 3, 5, -1, 5, 8, -2, 8, -1, 8, 13, -5, 1, 13, -3, 13, -2, 13, -1, 13, 21, -8, 1, 21, -8, 2, 21, -5, 21, -5, 1, 21, -3, 21, -2, 21, -1, 21, 34, -13, 1, 34, -13, 2, 34, -13, 3, 34, -13, 5, -1, 34, -8, 34, -8, 1, 34, -8, 2, 34, -5, 34, -5, 1, 34, -3, 34

Offset: 0

Clark Kimberling, Apr 08 2015

Suppose that b = (b(0), b(1), ... ) is an increasing sequence of positive integers satisfying b(0) = 1 and b(n+1) <= 2*b(n) for n >= 0. Let B(n) be the least b(m) >= n. Let R(0) = 1, and for n > 0, let R(n) = B(n) - R(B(n) - n). The resulting sum of the form R(n) = B(n) - B(m(1)) + B(m(2)) - ... + ((-1)^k)*B(k) is introduced here as the minimal alternating b-representation of n. The sum B(n) + B(m(2)) + ... we call the positive part of R(n), and the sum B(m(1)) + B(m(3)) + ... , the nonpositive part of R(n). The number ((-1)^k)*B(k) is the trace of n.

If b(n) = F(n+2), where F = A000045, then the sum is the minimal alternating Fibonacci-representation of n.

			R(0) = 0
R(1) = 1
R(2) = 2
R(3) = 3
R(4) = 5 - 1
R(9) = 13 - 5 + 1
R(25) = 34 - 13 + 5 - 1
R(64) = 89 - 34 + 13 - 5 + 1

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000045, A255973 (trace), A256656 (numbers with positive trace), A256657 (numbers with nonpositive trace), A256663 (positive part of R(n)), A256664 (nonpositive part of R(n)), A256654, A256696 (minimal alternating binary representations), A255974 (minimal alternating triangular-number representations), A256789 (minimal alternating squares representations).

f[n_] = Fibonacci[n]; ff = Table[f[n], {n, 1, 70}];
s[n_] := Table[f[n + 2], {k, 1, f[n]}];
h[0] = {1}; h[n_] := Join[h[n - 1], s[n]];
g = h[12]; r[0] = {0};
r[n_] := If[MemberQ[ff, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
Flatten[Table[r[n], {n, 0, 60}]]

R(F(k)^2) = F(2k-1) - F(2k-3) + F(2k-5) - ... + d*F(5) + (-1)^k, where d = (-1)^(k+1).

A087172 Greatest Fibonacci number that does not exceed n.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55

Offset: 1

Sam Alexander, Oct 19 2003

Also the largest term in Zeckendorf representation of n; starting at Fibonacci positions the sequence is repeated again and again in A107017: A107017(A000045(n)+k) = a(k) with 0 < k < A000045(n-1). - Reinhard Zumkeller, May 09 2005

Fibonacci(n) occurs Fibonacci(n-1) times, for n >= 2. - Benoit Cloitre, Dec 15 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A007895, A035516, A035517, A066628, A107017, A130233, A130234, A256654.

Partial sums: A130473.

a087172 = head . a035516_row -- Reinhard Zumkeller, Mar 10 2013

with(combinat):
A087172 := proc (n) local j: for j while fibonacci(j) <= n do fibonacci(j) end do: fibonacci(j-1) end proc:
seq(A087172(n), n = 1 .. 40); # Emeric Deutsch, Nov 11 2014
# Alternative
N:= 100: # to get a(n) for n from 1 to N
Fibs:= [seq(combinat:-fibonacci(i), i = 1 .. ceil(log[(1 + sqrt(5))/2](sqrt(5)*N)))]:
A:= Vector(N):
for i from 1 to nops(Fibs)-1 do
  A[Fibs[i] .. min(N,Fibs[i+1]-1)]:= Fibs[i]
od:
convert(A,list); # Robert Israel, Nov 11 2014

With[{rf=Reverse[Fibonacci[Range[10]]]},Flatten[Table[ Select[ rf,n>=#&, 1],{n,80}]]] (* Harvey P. Dale, Dec 08 2012 *)
Flatten[Map[ConstantArray[Fibonacci[#],Fibonacci[#-1]]&,Range[15]]] (* Peter J. C. Moses, May 02 2022 *)

a(n)=my(k=log(n)\log((1+sqrt(5))/2)); while(fibonacci(k)<=n, k++); fibonacci(k--) \\ Charles R Greathouse IV, Jul 24 2012

a(n) = Fibonacci(A130233(n)) = Fibonacci(A130234(n+1)-1). - Hieronymus Fischer, May 28 2007
a(n) = A035516(n, 0) = A035517(n, A007895(n)-1). - Reinhard Zumkeller, Mar 10 2013
a(n) = n - A066628(n). - Michel Marcus, Feb 02 2016
Sum_{n>=1} 1/a(n)^2 = Sum_{n>=1} Fibonacci(n)/Fibonacci(n+1)^2 = 1.7947486789... . - Amiram Eldar, Aug 16 2022

A255973 Trace of the minimal alternating Fibonacci representation of n.

Original entry on oeis.org

0, 1, 2, 3, -1, 5, -2, -1, 8, 1, -3, -2, -1, 13, 1, 2, -5, 1, -3, -2, -1, 21, 1, 2, 3, -1, -8, 1, 2, -5, 1, -3, -2, -1, 34, 1, 2, 3, -1, 5, -2, -1, -13, 1, 2, 3, -1, -8, 1, 2, -5, 1, -3, -2, -1, 55, 1, 2, 3, -1, 5, -2, -1, 8, 1, -3, -2, -1, -21, 1, 2, 3, -1

Offset: 0

Clark Kimberling, Apr 08 2015

See A256655 for definitions.

			Let R(k) be the minimal alternating Fibonacci representation of k.  The trace of R(k) is the last term.
R(1) = 1, trace = 1
R(2) = 2, trace = 2
R(3) = 3, trace = 3
R(4) = 5 - 1, trace = -1
R(5) = 5, trace = 5
R(6) = 6 - 2, trace = -2

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000045, A256655 (representations R(n)), A256656 (numbers with positive trace), A256657 (numbers with nonpositive trace), A256663 (positive part of R(n)), A256664 (nonpositive part of R(n)), A256654.

b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[12];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
Table[Last[r[n]], {n, 0, 200}]  (* A255973 *)

A333907 For n >= 1, a(n) = Sum_{k=1..n} prevfib(k) + nextfib(k) - 2*k, where prevfib(k) is the largest Fibonacci number < k, nextfib(k) is the smallest Fibonacci number > k.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 2, 4, 7, 8, 7, 4, 7, 13, 17, 19, 19, 17, 13, 7, 12, 23, 32, 39, 44, 47, 48, 47, 44, 39, 32, 23, 12, 20, 39, 56, 71, 84, 95, 104, 111, 116, 119, 120, 119, 116, 111, 104, 95, 84, 71, 56, 39, 20, 33, 65, 95, 123, 149, 173, 195, 215, 233, 249, 263, 275

Offset: 1

Ctibor O. Zizka, Apr 09 2020

nonn

			a(1) = (0 + 2 - 2*1) = 0;
a(2) = (0 + 2 - 2*1) + (1 + 3 - 2*2) = 0;
a(3) = (0 + 2 - 2*1) + (1 + 3 - 2*2) + (2 + 5 - 2*3) = 1;
a(4) = (0 + 2 - 2*1) + (1 + 3 - 2*2) + (2 + 5 - 2*3) + (3 + 5 - 2*4) = 1.

Cf. A000045, A001076, A087172, A130473, A194029, A256654, A280514.

isfib(k) = my(m=5*k^2); issquare(m-4) || issquare(m+4);
nextfib(n) = my(k=n+1); while (!isfib(k), k++); k;
prevfib(n) = my(k=n-1); while (!isfib(k), k--); k;
a(n) = sum(k=1, n, prevfib(k) + nextfib(k) - 2*k); \\ Michel Marcus, Apr 10 2020

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A256655 R(k), the minimal alternating Fibonacci representation of k, concatenated for k = 0, 1, 2,....

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A087172 Greatest Fibonacci number that does not exceed n.

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A255973 Trace of the minimal alternating Fibonacci representation of n.

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A333907 For n >= 1, a(n) = Sum_{k=1..n} prevfib(k) + nextfib(k) - 2*k, where prevfib(k) is the largest Fibonacci number < k, nextfib(k) is the smallest Fibonacci number > k.

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