A254123 Least number of terms needed to build n from Fibonacci numbers using + and *.

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

Offset: 1

Author

Robert Israel, Jan 25 2015

Keywords

• nonn

Comments

a(n) = 1 iff n is a Fibonacci number (A000045).

a(n) = 2 iff n is in A254122.

a(A025282(n)) = n.

Examples

			a(40) = 2 because 40 = 5*8; 5 and 8 are Fibonacci numbers but 40 is not.
		

Links

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

Crossrefs

Cf. A000045, A025282, A254122.

Programs

Maple

N:= 1000: # to get a(1) to a(N)
A:= Vector(N):
for i from 1 do
  f:= combinat:-fibonacci(i);
  if f > N then break fi;
  A[f]:= 1
od:
for n from 1 to N do
  if A[n] = 0 then
     m:= floor(n/2);
     r:= min(A[1..m] + A[[seq(n-i,i=1..m)]]);
     for a in select(`<=`, numtheory:-divisors(n) minus {1}, floor(sqrt(n))) do
        r:= min(r, A[a] + A[n/a])
     od:
     A[n]:= r;
  fi
od:
seq(A[i],i=1..N);