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A386743 For n >= 0, a(n) is the least Fibonacci number F(i) such that F(i) = (2*n + 1)*(n + k) for some k >= 0.

%I A386743 #17 Aug 12 2025 02:40:14
%S A386743 0,3,55,21,144,55,377,6765,2584,2584,987,46368,75025,14930352,317811,
%T A386743 832040,6765,102334155,4181,317811,6765,701408733,1548008755920,
%U A386743 2178309,225851433717,14930352,196418,6765,14930352,591286729879,832040,46368,9227465,72723460248141,46368
%N A386743 For n >= 0, a(n) is the least Fibonacci number F(i) such that F(i) = (2*n + 1)*(n + k) for some k >= 0.
%C A386743 a(n) >= n*(2*n + 1).
%C A386743 Empirical observation: For F(i), i is from {1/3, 1/2, 2/3, 1, 3/2, 2}*A001177(2*n + 1).
%e A386743 For n = 3: F(i) = 21 + 7*k is true for k = 0, F(8) = 21, thus a(3) = 21.
%t A386743 a[n_]:=Module[{i=0},While[!Divisible[Fibonacci[i],2n+1] || Fibonacci[i]/(2n+1) < n, i++]; Fibonacci[i]]; Array[a,35,0] (* _Stefano Spezia_, Aug 04 2025 *)
%o A386743 (PARI) a(n) = my(i=0, f=fibonacci(0)); while((f % (2*n+1)) || (f/(2*n+1) < n), i++; f=fibonacci(i)); f; \\ _Michel Marcus_, Aug 01 2025
%o A386743 (Python)
%o A386743 def A386743(n):
%o A386743     a, b, m, k = 0, 1, n*((n<<1)|1), (n<<1)|1
%o A386743     while a<m or a%k:
%o A386743         a, b = b, a+b
%o A386743     return a # _Chai Wah Wu_, Aug 11 2025
%Y A386743 Cf. A000045, A001177.
%K A386743 nonn
%O A386743 0,2
%A A386743 _Ctibor O. Zizka_, Aug 01 2025
%E A386743 More terms from _Michel Marcus_, Aug 01 2025