A130473 -id:A130473 - cp's OEIS frontend

A087172 Greatest Fibonacci number that does not exceed n.

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

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

A351628 Partial sums of A352717.

Original entry on oeis.org

1, 2, 5, 9, 13, 17, 24, 31, 38, 45, 56, 67, 78, 89, 100, 111, 122, 140, 158, 176, 194, 212, 230, 248, 266, 284, 302, 320, 349, 378, 407, 436, 465, 494, 523, 552, 581, 610, 639, 668, 697, 726, 755, 784, 813, 842, 889, 936, 983, 1030, 1077, 1124, 1171, 1218

Offset: 1

Clark Kimberling, May 04 2022

nonn

Cf. A087172, A130473, A353717.

Accumulate[Flatten[Map[ConstantArray[LucasL[#],LucasL[#-1]]&,Range[15]]]] (* Peter J. C. Moses, May 02 2022 *)

from itertools import islice
def A351628_gen(): # generator of terms
    a, b, c = 1, 3, 0
    while True:
        yield from (c+i*a for i in range(1,b-a+1))
        a, b, c = b, a+b, c + a*(b-a)
A351628_list = list(islice(A351628_gen(),40)) # Chai Wah Wu, Jun 09 2022

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A087172 Greatest Fibonacci number that does not exceed 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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A351628 Partial sums of A352717.

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