A364353 Numbers that cannot be expressed as the sum of the squares of four Fibonacci numbers.

24, 32, 40, 41, 45, 46, 48, 49, 53, 56, 57, 61, 62, 71, 80, 85, 87, 88, 92, 95, 96, 101, 103, 104, 105, 106, 108, 109, 110, 111, 112, 113, 116, 117, 119, 120, 121, 122, 124, 125, 126, 127, 131, 134, 135, 140, 142, 143, 144

Offset

1,1

Comments

User vvg asked on Math StackExchange if this sequence is empty, by analogy with Lagrange's Four Square Theorem and Zeckendorf's Theorem (see links). As mentioned by user Empy2, 'most' of the positive integers belong to this sequence: there are O(log n) Fibonacci squares below n, so there are at most O( (log n)^4 ) distinct sums below n, which is asymptotically much smaller than n.

Links

Table of n, a(n) for n=1..49.

Math StackExchange, Is every integer representable as the sum of four Fibonacci squares?.

Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem.

Eric Weisstein's World of Mathematics, Zeckendorf's Theorem.

Example

24 is a term, which can be seen as follows.  The Fibonacci squares less than 24 are 0,1,4 and 9. We must use at least two 9s, as 24>9+4+4+4=21. But 24-18=6 is neither a square nor a sum of two Fibonacci squares. Hence, 24 has no representation of the required form.
32 is a term, as follows. The Fibonacci squares less than 32 are 0,1,4,9, and 25. We cannot use 25 as 7 cannot be written with 3 Fibonacci squares. Therefore, we must have at least three 9s, but 32-9-9-9=5 is not a Fibonacci square. The result follows.

Prog

(Scala)
@main def main(args: Int*): Unit =
    val max = 10000
    def fibFrom(n: Int, m: Int): LazyList[Int] =
      n #:: m #:: fibFrom(n + m, n + 2 * m)
    val fib = fibFrom(0, 1)
    def computeReprs(min: Int, max: Int) = (for
      n <- min to max
      f1 <- fib.takeWhile(f => f * f <= n)
      f2 <- fib.dropWhile(_ < f1).takeWhile(f => f * f <= n - f1 * f1)
      f3 <- fib
        .dropWhile(_ < f2)
        .takeWhile(f => f * f <= n - f1 * f1 - f2 * f2)
      f4 = fib
        .dropWhile(f => f * f < n - f1 * f1 - f2 * f2 - f3 * f3)
        .head
      if n == f1 * f1 + f2 * f2 + f3 * f3 + f4 * f4
    yield (n +: Seq(f1, f2, f3, f4)
      .filter(_ != 0)
      .map(fib.indexOf(_)))).distinct
  val reprs = computeReprs(0, max)
    // printing out to console for creating b-file
  val nosWithoutReprs = (0 to max)
      .filter(i => reprs.forall(k => k(0) != i))
  ((1 to nosWithoutReprs.length) zip nosWithoutReprs)
    .foreach((p, q) => println(s"$p $q"))

Crossrefs

Cf. A000045, A007598, A364354 (complement).

Sequence in context: A334589 A334936 A102374 * A317534 A240068 A269424

Adjacent sequences: A364350 A364351 A364352 * A364354 A364355 A364356

Keyword

nonn

Author

Calvin Khor, Jul 20 2023

Status

approved