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
Examples
17 = 16+1, 16 is a square and 1 is a Fibonacci number, so 17 is not in the sequence.
Programs
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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 *) -
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
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