A377536 Integers that are the arithmetic mean of two distinct Fibonacci numbers (A000045).

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 28, 29, 30, 34, 38, 45, 46, 47, 51, 55, 72, 73, 76, 89, 117, 118, 119, 123, 127, 144, 161, 189, 190, 191, 195, 199, 216, 233, 305, 306, 309, 322, 377, 494, 495, 496, 500, 504, 521, 538, 610, 682, 799, 800, 801, 805

Offset: 1

Author

Felix Huber, Dec 18 2024

Keywords

• nonn

Comments

This sequence contains all positive Fibonacci numbers of A000045. Proof: For i >= 2, (F(i-2) + F(i+1))/2 = (F(i-2) + F(i-1) + F(i))/2 = (F(i-2) + F(i-1) + F(i-2) + F(i-1))/2 = F(i-1) + F(i-2) = F(i).

Examples

			1 is in the sequence because (F(0) + F(3))/2 = (0 + 2)/2 = 1.
12 is in the sequence because (F(4) + F(8))/2 = (3 + 21)/2 = 12.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fibonacci Number

Crossrefs

Cf. A000045, A084176.

Programs

Maple

with(combinat):
A377536:=proc(k)
   local L,M,i,j;
   M:={};
   L:=[seq(fibonacci(i),i=0..k)];
   for i to k do
      for j from i+1 to k+1 do
         if is(L[i]+L[j],even) then
            M:=[op(M),(L[i]+L[j])/2]
         fi
      od
   od;
   M:=convert(M,set);
   return op(M)
end proc:
A377536(17)