Tetranacci and pentanacci numbers

DefinitionPentanacci numbers#✓fully proved

The pentanacci numbers \(a_n\), for natural numbers \(n\), are defined by \(a_0 = 1\), \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 4\), \(a_4 = 8\) and \(a_{n+5} = a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_n\) for every \(n\), so each term from index \(5\) on is the sum of the five terms before it.

Their ordinary generating function is \(1/(1 - t - t^2 - t^3 - t^4 - t^5)\); up to a shift of index they form OEIS A001591.

def pentanacciNumber : ℕ → ℕ
  | 0 => 1
  | 1 => 1
  | 2 => 2
  | 3 => 4
  | 4 => 8
  | n + 5 =>
      pentanacciNumber (n + 4) + pentanacciNumber (n + 3) +
        pentanacciNumber (n + 2) + pentanacciNumber (n + 1) +
        pentanacciNumber n

DefinitionTetranacci sequence#✓fully proved

The Tetranacci sequence 0, 0, 0, 1, 1, 2, 4, 8, … (OEIS A000078), in which each term after the first four is the sum of its four predecessors.

def tetranacci : ℕ → ℕ
  | 0 => 0
  | 1 => 0
  | 2 => 0
  | 3 => 1
  | n + 4 =>
      tetranacci (n + 3) + tetranacci (n + 2) + tetranacci (n + 1) + tetranacci n

Used by 1

Used byThe Brocard–Ramanujan equation has no Tetranacci solutions

2 fully proved

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