Smarandache permutation sequence

DefinitionBlocks of the Smarandache permutation sequence#✓fully proved

For a natural number \(n\), the \(n\)-th block is the list of the odd numbers \(1, 3, \ldots, 2n - 1\) in increasing order followed by the even numbers \(2n, 2n - 2, \ldots, 2\) in decreasing order; the \(0\)-th block is empty.

def smarandacheBlock (n : ℕ) : List ℕ :=
  ((List.range n).map (fun k => 2 * k + 1)) ++ ((List.range n).map (fun k => 2 * (n - k)))

Used by 5

Used byFourth block of the Smarandache permutation sequence · First block of the Smarandache permutation sequence · Third block of the Smarandache permutation sequence · Second block of the Smarandache permutation sequence · Prefix of the Smarandache permutation sequence

DefinitionPrefix of the Smarandache permutation sequence#✓fully proved

For a natural number \(N\), this is the list of natural numbers obtained by concatenating, in order, the Smarandache blocks numbered \(1, 2, \dots, N\), where block \(n\) is the list \((1, 3, \dots, 2n - 1, 2n, 2n - 2, \dots, 2)\) of the odd numbers below \(2n\) in increasing order followed by the even numbers from \(2n\) down to \(2\). For \(N = 0\) it is the empty list.

def smarandachePrefix (N : ℕ) : List ℕ :=
  ((List.range N).map (fun n => smarandacheBlock (n + 1))).flatten

Uses 1 · Used by 2

Statement usesBlocks of the Smarandache permutation sequence

Used bySmarandache permutation sequence · Concatenation of the first three Smarandache blocks

DefinitionSmarandache permutation sequence#✓fully proved

The Smarandache permutation sequence (OEIS A004741) is the sequence \((a_i)_{i \ge 0}\) of natural numbers, indexed from \(0\), obtained by concatenating the blocks \((1, 3, \ldots, 2n - 1, 2n, 2n - 2, \ldots, 2)\) for \(n = 1, 2, 3, \ldots\) in order; it begins \(1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2, \ldots\), so \(a_0 = 1\).

def smarandachePermutation (i : ℕ) : ℕ :=
  (smarandachePrefix (i + 1)).getD i 0

Uses 1 · Used by 3

Statement usesPrefix of the Smarandache permutation sequence

Used byTwelfth term of the Smarandache permutation sequence · Sixth term of the Smarandache permutation sequence · Initial term of the Smarandache permutation sequence

TheoremFirst block of the Smarandache permutation sequence#✓fully proved

The first block of the Smarandache permutation sequence is \((1, 2)\).

theorem smarandacheBlock_one : smarandacheBlock 1 = [1, 2]

Uses 1

Statement usesBlocks of the Smarandache permutation sequence

TheoremSecond block of the Smarandache permutation sequence#✓fully proved

The second block of the Smarandache permutation sequence is \((1, 3, 4, 2)\).

theorem smarandacheBlock_two : smarandacheBlock 2 = [1, 3, 4, 2]

Uses 1

Statement usesBlocks of the Smarandache permutation sequence

TheoremThird block of the Smarandache permutation sequence#✓fully proved

The third block of the Smarandache permutation sequence is \((1, 3, 5, 6, 4, 2)\).

theorem smarandacheBlock_three : smarandacheBlock 3 = [1, 3, 5, 6, 4, 2]

Uses 1

Statement usesBlocks of the Smarandache permutation sequence

TheoremFourth block of the Smarandache permutation sequence#✓fully proved

The fourth block of the Smarandache permutation sequence is \((1, 3, 5, 7, 8, 6, 4, 2)\).

theorem smarandacheBlock_four :
    smarandacheBlock 4 = [1, 3, 5, 7, 8, 6, 4, 2]

Uses 1

Statement usesBlocks of the Smarandache permutation sequence

TheoremConcatenation of the first three Smarandache blocks#✓fully proved

Concatenating the first three blocks of the Smarandache permutation sequence gives \((1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2)\).

theorem smarandachePrefix_three :
    smarandachePrefix 3 = [1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2]

Uses 1

Statement usesPrefix of the Smarandache permutation sequence

TheoremInitial term of the Smarandache permutation sequence#✓fully proved

Indexing the Smarandache permutation sequence \((a_i)_{i \ge 0}\) from \(0\), its initial term is \(a_0 = 1\).

theorem smarandachePermutation_zero : smarandachePermutation 0 = 1

Uses 1

Statement usesSmarandache permutation sequence

TheoremSixth term of the Smarandache permutation sequence#✓fully proved

Indexing the Smarandache permutation sequence \((a_i)_{i \ge 0}\) from \(0\), its sixth term is \(a_5 = 2\).

theorem smarandachePermutation_five : smarandachePermutation 5 = 2

Uses 1

Statement usesSmarandache permutation sequence

TheoremTwelfth term of the Smarandache permutation sequence#✓fully proved

Indexing the Smarandache permutation sequence \((a_i)_{i \ge 0}\) from \(0\), its twelfth term is \(a_{11} = 2\).

theorem smarandachePermutation_eleven : smarandachePermutation 11 = 2

Uses 1

Statement usesSmarandache permutation sequence

11 fully proved

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