A214410 Numbers that can't be expressed as the sum of a square and a Fibonacci number.

15, 20, 23, 31, 32, 40, 42, 45, 47, 48, 53, 58, 60, 61, 63, 68, 73, 74, 75, 76, 78, 79, 87, 88, 92, 95, 96, 97, 99, 106, 107, 109, 110, 111, 112, 116, 117, 118, 120, 127, 128, 130, 131, 132, 133, 135, 137, 139, 140, 141, 143, 150, 151, 154, 156, 158, 159, 161

Offset

1,1

Comments

0 is considered to be a Fibonacci number.

Links

Table of n, a(n) for n=1..58.

Example

17 = 16+1, 16 is a square and 1 is a Fibonacci number, so 17 is not in the sequence.

Maple

q:= proc(n) local f, g; f, g:= 0, 1;
      do if f>n        then return true
       elif issqr(n-f) then return false
       else f, g:= g, f+g
      fi od
    end:
select(q, [$0..200])[];  # Alois P. Heinz, May 22 2021

Mathematica

nn = 161; sq = Range[0, Sqrt[nn]]^2; fb = {}; i = 0; While[f = Fibonacci[i];  f < nn, i++; AppendTo[fb, f]]; fb = Union[fb]; Complement[Range[0, nn], Union[Flatten[Outer[Plus, sq, fb]]]] (* T. D. Noe, Jul 31 2012 *)

Prog

(Python)
prpr = 0
prev = 1
fib = [0]*100
for n in range(100):
    fib[n] = prpr
    curr = prpr+prev
    prpr = prev
    prev = curr
for n in range(1234):
    i = yes = 0
    while i*i<=n:
        r = n - i*i
        if r in fib:
            yes = 1
            break
        i += 1
    if yes==0:
        print(n, end=', ')
(Python)
from sympy import fibonacci
from itertools import count, takewhile
def aupto(lim):
  fbs = list(takewhile(lambda x: x<=lim, (fibonacci(i) for i in count(0))))
  sqs = list(takewhile(lambda x: x<=lim, (i*i for i in count(0))))
  return sorted(set(range(1, lim+1)) - set(f+s for f in fbs for s in sqs))
print(aupto(161)) # Michael S. Branicky, May 22 2021

Crossrefs

Cf. A000045, A000290, A132144, A214412.

Sequence in context: A171808 A134617 A113529 * A102495 A133288 A322710

Adjacent sequences: A214407 A214408 A214409 * A214411 A214412 A214413

Keyword

nonn,easy

Author

Alex Ratushnyak, Jul 16 2012

Status

approved