A108852 -id:A108852 - cp's OEIS frontend

A072649 n occurs Fibonacci(n) times (cf. A000045).

1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Antti Karttunen, Jun 02 2002

Number of digits in Zeckendorf-binary representation of n. E.g., the Zeckendorf representation of 12 is 8+3+1, which in binary notation is 10101, which consists of 5 digits. - Clark Kimberling, Jun 05 2004
First position where value n occurs is A000045(n+1), i.e., a(A000045(n)) = n-1, for n >= 2 and a(A000045(n)-1) = n-2, for n >= 3.
This is the number of distinct Fibonacci numbers greater than 0 which are less than or equal to n. - Robert G. Wilson v, Dec 10 2006
The smallest nondecreasing sequence a(n) such that a(Fibonacci(n-1)) = n. - Tanya Khovanova, Jun 20 2007

			1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 4.
Popular Computing (Calabasas, CA), A Coding Exercise (from a suggestion by R. W. Hamming), Vol. 5 (No. 54, Sep 1977), p. PC55-18.

Cf. A000045, A095791, A095792.
Cf. A001622 (golden ratio phi), A073010.
Used to construct A003714. Cf. also A002024, A072643, A072648, A072650.
Cf. A131234.
Partial sums: A256966, A256967.

a072649 n = a072649_list !! (n-1)
a072649_list = f 1 where
   f n = (replicate (fromInteger $ a000045 n) n) ++ f (n+1)
-- Reinhard Zumkeller, Jul 04 2011

A072649 := proc(n)
    local j;
    for j from ilog[(1+sqrt(5))/2](n) while combinat[fibonacci](j+1)<=n do
    end do;
    j-1
end proc:
seq(A072649(n), n=1..120);  # Alois P. Heinz, Mar 18 2013

Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* Robert G. Wilson v, Jan 14 2007 *)
a[n_] := Module[{j}, For[j = Floor@Log[GoldenRatio, n], Fibonacci[j+1] <= n, j++]; j-1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)

a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2))

a(n)=local(m); if(n<1,0,m=0; until(fibonacci(m)>n,m++); m-2)

from sympy import fibonacci
def a(n):
    if n<1: return 0
    m=0
    while fibonacci(m)<=n: m+=1
    return m-2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 09 2017

def A072649(n):
    a, b, c = 0, 1, -2
    while a <= n:
        a, b = b, a+b
        c += 1
    return c # Chai Wah Wu, Nov 04 2024
(MIT/GNU Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) ;; (The implementation below is better)

(define (A072649 n) (if (<= n 3) n (let loop ((k 5)) (if (> (A000045 k) n) (- k 2) (loop (+ 1 k)))))) ;; (Use this with the memoized implementation of A000045 given under that entry. No floating point arithmetic is involved). - Antti Karttunen, Oct 06 2017

G.f.: (Sum_{n>1} x^Fibonacci(n))/(1-x). - Michael Somos, Apr 25 2003
From Hieronymus Fischer, May 02 2007: (Start)
a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) - 1, where phi is A001622.
a(n) = floor(arcsinh(sqrt(5)*n/2)/log(phi)) - 1.
a(n) = A108852(n) - 2. (End)
a(n) = -1 + floor( log_phi( (n+0.2)*sqrt(5) ) ), where log_phi(x) is the logarithm to the base (1+sqrt(5))/2. - Ralf Stephan, May 14 2007
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 18 2024

Typo fixed by Charles R Greathouse IV, Oct 28 2009

A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Understood as a binary number, Sum_{k>=0} a(k)/2^k, the resulting decimal expansion is 1.910278797207865891... = Fibonacci_binary+0.5 (see A084119) or Fibonacci_binary_constant-0.5 (see A124091), respectively. - Hieronymus Fischer, May 14 2007
a(n)=1 if and only if there is an integer m such that x=n is a root of p(x)=25*x^4-10*m^2*x^2+m^4-16. Also a(n)=1 iff floor(s)<>floor(c) or ceiling(s)<>ceiling(c) where s=arcsinh(sqrt(5)*n/2)/log(phi), c=arccosh(sqrt(5)*n/2)/log(phi) and phi=(1+sqrt(5))/2. - Hieronymus Fischer, May 17 2007
a(A000045(n)) = 1; a(A001690(n)) = 0. - Reinhard Zumkeller, Oct 10 2013
Image, under the map sending a,b,c -> 1, d,e,f -> 0, of the fixed point, starting with a, of the morphism sending a -> ab, b -> c, c -> cd, d -> d, e -> ef, f -> e. - Jeffrey Shallit, May 14 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux (2004), Volume: 16, Issue: 3, page 487-518.
Wikipedia, Fibonacci number
Index entries for characteristic functions

Cf. A000045, A001690, A072649, A104162, A108852, A124091, A130233, A130234.
Decimal expansion of Fibonacci binary is in A084119.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
Cf. A079586 (Dirich. g.f. at s=1).

import Data.List (genericIndex)
a010056 = genericIndex a010056_list
a010056_list = 1 : 1 : ch [2..] (drop 3 a000045_list) where
   ch (x:xs) fs'@(f:fs) = if x == f then 1 : ch xs fs else 0 : ch xs fs'
-- Reinhard Zumkeller, Oct 10 2013

a:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
seq(a(n), n=0..144);  # Alois P. Heinz, Dec 06 2020

Join[{1},With[{fibs=Fibonacci[Range[15]]},If[MemberQ[fibs,#],1,0]& /@Range[100]]]  (* Harvey P. Dale, May 02 2011 *)

a(n)=my(k=n^2);k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)) \\ Charles R Greathouse IV, Jul 30 2012

from sympy.ntheory.primetest import is_square
def A010056(n): return int(is_square(m:=5*n**2-4) or is_square(m+8)) # Chai Wah Wu, Mar 30 2023

G.f.: (Sum_{k>=0} x^A000045(k)) - x. - Hieronymus Fischer, May 17 2007

A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Hieronymus Fischer, May 17 2007

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 1 (see A130234 for another version). a(n) + 1 is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

			a(10) = 6, since Fibonacci(6) = 8 <= 10 but Fibonacci(7) = 13 > 10.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A130235 (partial sums), A104162 (first differences).
Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A108852. Lucas inverse: A130241.
Cf. A001622 (golden ratio), A002390 (its log).

fibLLog[0] := 0; fibLLog[1] := 2; fibLLog[n_Integer] := fibLLog[n] = If[n < Fibonacci[fibLLog[n - 1] + 1], fibLLog[n - 1], fibLLog[n - 1] + 1]; Table[fibLLog[n], {n, 0, 88}] (* Alonso del Arte, Sep 01 2013 *)

a(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2) \\ Charles R Greathouse IV, Mar 21 2012

a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) where phi = (1+sqrt(5))/2 = A001622.
a(n) = floor(arcsinh(sqrt(5)*n/2) / log(phi)), with log(phi) = A002390.
a(n) = A130234(n+1) - 1.
G.f.: g(x) = 1/(1-x) * Sum_{k>=1} x^Fibonacci(k).
a(n) = floor(log_phi(sqrt(5)*n+1)), n >= 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007

A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).

Original entry on oeis.org

0, 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Hieronymus Fischer, May 17 2007

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 2 (see A130233 for another version). a(n+1) is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

			a(10) = 7, since Fibonacci(7) = 13 >= 10 but Fibonacci(6) = 8 < 10.

Partial sums: A130236.
Other related sequences: A000045, A130233, A130256, A130260, A104162, A108852.
Lucas inverse: A130241 - A130248.

A130234 := proc(n)
    local i;
    for i from 0 do
        if A000045(i) >= n then
            return i;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 31 2015

a[n_] := For[i = 0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
a /@ Range[0, 80] (* Jean-François Alcover, Apr 13 2020 *)

a(n)=my(k);while(fibonacci(k)Charles R Greathouse IV, Feb 03 2014, corrected by M. F. Hasler, Apr 07 2021

a(n) = ceiling(log_phi((sqrt(5)*n + sqrt(5*n^2-4))/2)) = ceiling(arccosh(sqrt(5)*n/2)/log(phi)) where phi = (1+sqrt(5))/2, the golden ratio, for n > 0.
a(n) = A130233(n-1) + 1 for n > 0.
G.f.: x/(1-x) * Sum_{k >= 0} x^Fibonacci(k).
a(n) = ceiling(log_phi(sqrt(5)*n - 1)) for n > 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007
a(n) = A108852(n-1). - R. J. Mathar, Jan 31 2015

A104162 Indicator sequence for the Fibonacci numbers.

Original entry on oeis.org

1, 2, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Apr 01 2005

Without multiplicities, this is A010056.

The number of nonnegative integer solutions of x^4 - 10*n^2*x^2 + 25*n^4 - 16 = 0. - Hieronymus Fischer, May 17 2007

			a(1)=2 since F(1)=F(2)=1.

Cf. A000045.
Partial sums are in A108852.
See also A130233 and A130234.

a(n)=if(n==1,return(2)); my(k=n^2);k+=((k + 1) << 2);issquare(k) || issquare(k-8) \\ Charles R Greathouse IV, Feb 03 2014; typo corrected by Georg Fischer, Jun 22 2022

G.f.: Sum_{k>=0} x^Fibonacci(k).
From Hieronymus Fischer, May 17 2007: (Start)
a(n) = 1+floor(arcsinh(sqrt(5)*n/2)/log(phi))-ceiling(arccosh(sqrt(5)*n/2)/log(phi)), for n>0, where phi=(1+sqrt(5))/2.
a(n) = A108852(n) - A108852(n-1) for n>0.
a(n) = A130233(n) - A130233(n-1) for n>0.
a(n) = 1 + A130233(n) - A130234(n) for n>0.
a(n) = A130234(n+1) - A130234(n) for n>=0. (End)

A130245 Number of Lucas numbers (A000032) <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Partial sums of the Lucas indicator sequence A102460.

For n>=2, we have a(A000032(n)) = n + 1.

			a(9)=5 because there are 5 Lucas numbers <=9 (2,1,3,4 and 7).

Antti Karttunen, Table of n, a(n) for n = 0..64079
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Partial sums of A102460.
For partial sums of this sequence, see A130246. Other related sequences: A000032, A130241, A130242, A130247, A130249, A130253, A130255, A130259.
For Fibonacci inverse, see A130233 - A130240, A104162, A108852.

[0] cat [1+Floor(Log((2*n+1)/2)/Log((1+Sqrt(5))/2)): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Join[{0}, Table[1+Floor[Log[GoldenRatio, (2*n+1)/2]], {n,1,100}]] (* G. C. Greubel, Sep 09 2018 *)

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A130245(n) = if(!n,n,A102460(n)+A130245(n-1));
\\ Or just as:
c=0; for(n=0,123,c += A102460(n); print1(c,", ")); \\ Antti Karttunen, May 13 2018

from itertools import count, islice
def A130245_gen(): # generator of terms
    yield from (0, 1, 2)
    a, b = 3,4
    for i in count(3):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130245_list = list(islice(A130245_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = 1 +floor(log_phi((n+sqrt(n^2+4))/2)) = 1 +floor(arcsinh(n/2)/log(phi)) for n>=2, where phi = (1+sqrt(5))/2.
a(n) = A130241(n)+1 = A130242(n+1) for n>=2.
G.f.: g(x) = 1/(1-x)*sum{k>=0, x^Lucas(k)}.
a(n) = 1 +floor(log_phi(n+1/2)) for n>=1, where phi is the golden ratio.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - Pi/(6*sqrt(3)) - log(3)/2. - Amiram Eldar, Jul 25 2025

A130253 Number of Jacobsthal numbers (A001045) <=n.

Original entry on oeis.org

1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Hieronymus Fischer, May 20 2007

Partial sums of the Jacobsthal indicator sequence (A105348).

For n<>1, we have a(A001045(n))=n+1.

			a(9)=5 because there are 5 Jacobsthal numbers <=9 (0,1,1,3 and 5).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

For partial sums see A130252. Other related sequences A001045, A130249, A130250, A130253, A105348. Also A130233, A130235, A130241, A108852, A130245.

[Ceiling(Log(3*n+2)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[1+Floor[Log[2,3n+1]],{n,0,100}] (* Harvey P. Dale, Jul 03 2013 *)

a(n)=logint(3*n+1,2)+1 \\ Charles R Greathouse IV, Oct 03 2016

def A130253(n): return (3*n+1).bit_length() # Chai Wah Wu, Apr 17 2025

a(n) = floor(log_2(3n+1)) + 1 = ceiling(log_2(3n+2)).
a(n) = A130249(n) + 1 = A130250(n+1).
G.f.: 1/(1-x)*(Sum_{k>=0} x^A001045(k)).

A362503 a(n) is the number of k such that n - A000045(k) is a square.

Original entry on oeis.org

1, 3, 3, 2, 2, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 0, 1, 4, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 3, 3, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 3, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 0, 1, 0, 2, 2, 1, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 2

Offset: 0

Robert Israel, Apr 22 2023

nonn

Number of ways to write n as the sum of a Fibonacci number and a square, where A000045(1) and A000045(2) are counted as separate.

			a(5) = 3 because 5 = A000045(1) + 2^2 = A000045(2) + 2^2 = A000045(5) + 0^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A108852, A362409, A362434.

N:= 100: # to get terms <= N
V:= Array(0..N):
for i from 0 do
  f:= combinat:-fibonacci(i);
  if f >= N then break fi;
  s:= floor(sqrt(N-f));
  J:=[seq(f+i^2, i=0..s)];
  V[J]:= V[J] +~ 1;
od:
convert(V,list);

f(n) = my(k=1); while (fibonacci(k) <= n, k++); k; \\ A108852
a(n) = sum(k=0, f(n), issquare(n-fibonacci(k))); \\ Michel Marcus, Apr 23 2023

a(1 + A000045(6*k)^2/4) >= 4.

A375642 a(n) is the number of i for which n - Fibonacci(i) is prime.

Original entry on oeis.org

0, 1, 3, 3, 3, 3, 3, 4, 1, 3, 2, 3, 3, 3, 3, 3, 1, 4, 3, 4, 2, 2, 2, 5, 2, 3, 1, 2, 1, 3, 3, 5, 1, 3, 0, 3, 3, 3, 3, 2, 2, 4, 2, 5, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 1, 4, 3, 5, 2, 3, 1, 3, 2, 4, 2, 1, 2, 5, 2, 6, 3, 2, 1, 2, 2, 4, 3, 2, 1, 5, 1, 3, 2, 2, 1, 2, 3, 5, 1, 3, 1, 3, 2, 3, 1

Offset: 1

Robert Israel, Aug 22 2024

nonn

			a(5) = 3 because 5 - Fibonacci(0) = 5, 5 - Fibonacci(3) = 3 and 5 - Fibonacci(4) = 2 are prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A108852, A132144, A168383, A169791.

fcount:= proc(n) local f,i,d,c;
  c:= 0;
  for i from 0 do
    f:= combinat:-fibonacci(i);
    if f >= n then return c fi;
    if isprime(n-f) then
      c:= c+1;
    fi
  od;
end proc:
map(f, [$1..200]);

a[n_]:=Sum[Boole[PrimeQ[n-Fibonacci[i]]],{i,Select[Range[0,n],n>Fibonacci[#]&]}]; Array[a,99] (* Stefano Spezia, Aug 23 2024 *)

A335741 Number of Pell numbers (A000129) <= n.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Ovidiu Bagdasar, Jun 20 2020

nonn

The sequence is constant on the interval A000129(k) < n <= A000129(k+1).

			The Pell numbers A000129 are 0,1,2,5,12,29,70,...
We have a(2)=a(3)=a(4)=3, since there are three Pell numbers less than or equal to 2,3 and 4, respectively.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Cf. A000129 (Pell Numbers), A108852 (Fibonacci), A130245 (Lucas), A130253 (Jacobsthal).

Partial sums of A105349.

Block[{a = 2, b = -1, nn = 105, u, v = {}}, u = {0, 1}; Do[AppendTo[u, Total[{-b, a} u[[-2 ;; -1]]]]; AppendTo[v, Count[u, ?(# <= i &)]], {i, nn}]; {Boole[First[u] <= 0]}~Join~v] (* _Michael De Vlieger, Jun 11 2021 *)
Module[{pn=LinearRecurrence[{2,1},{0,1},9],nn=100},Accumulate[Table[If[ MemberQ[ pn,n],1,0],{n,0,nn}]]] (* Harvey P. Dale, Apr 10 2022 *)

a(n) = 1+floor(log_alpha(2*sqrt(2)*n+1)), n>=0, where alpha=1+sqrt(2).

cp's OEIS Frontend

A072649 n occurs Fibonacci(n) times (cf. A000045).

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A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

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A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).

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A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).

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A104162 Indicator sequence for the Fibonacci numbers.

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A130245 Number of Lucas numbers (A000032) <= n.

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