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

%I A377536 #8 Dec 23 2024 22:18:25
%S A377536 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,
%T A377536 73,76,89,117,118,119,123,127,144,161,189,190,191,195,199,216,233,305,
%U A377536 306,309,322,377,494,495,496,500,504,521,538,610,682,799,800,801,805
%N A377536 Integers that are the arithmetic mean of two distinct Fibonacci numbers (A000045).
%C A377536 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).
%H A377536 Felix Huber, <a href="/A377536/b377536.txt">Table of n, a(n) for n = 1..10000</a>
%H A377536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%e A377536 1 is in the sequence because (F(0) + F(3))/2 = (0 + 2)/2 = 1.
%e A377536 12 is in the sequence because (F(4) + F(8))/2 = (3 + 21)/2 = 12.
%p A377536 with(combinat):
%p A377536 A377536:=proc(k)
%p A377536    local L,M,i,j;
%p A377536    M:={};
%p A377536    L:=[seq(fibonacci(i),i=0..k)];
%p A377536    for i to k do
%p A377536       for j from i+1 to k+1 do
%p A377536          if is(L[i]+L[j],even) then
%p A377536             M:=[op(M),(L[i]+L[j])/2]
%p A377536          fi
%p A377536       od
%p A377536    od;
%p A377536    M:=convert(M,set);
%p A377536    return op(M)
%p A377536 end proc:
%p A377536 A377536(17)
%Y A377536 Cf. A000045, A084176.
%K A377536 nonn
%O A377536 1,2
%A A377536 _Felix Huber_, Dec 18 2024