A282283 Recursive 2-parameter sequence allowing calculation of the Euler Totient function.

0, 1, -1, 1, 2, -4, 2, -4, 10, -6, -2, 2, 6, -16, 10, 4, -6, 8, -10, 4, -10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2, 12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6, -18, 52, -34, -16, 18, -18, 34, -36, 20, 10, -6, -2, 4, -28, 18, 8

Offset

0,5

Comments

The a(n,m) forms a table where each row has (n*(n-3)+4)/2 = A152947(n) elements.

The index of the first row is n=1 and the index of the first column is m=0.

The right diagonal a(n, A152947(n)) = A000010(n), Euler Totient function.

Links

Table of n, a(n) for n=0..64.

Formula

nu(n) = (n*(n-3)+4)/2

Q(n,m) = 2*A231599(n,m-1)-A231599(n,m-2)-A231599(n,m)

a(n, m) = a(n - 1, m - n + 1) - a(n - 1, m) - a(n - 1, nu(n - 1))*Q(n - 1, m) if (m < 0) or (nu(n) < m)

a(1,m)=1 if m=1 and 0 otherwise.

a(n,nu(n))= A000010(n)

Example

The first few rows are:
0, 1;
-1, 1;
2, -4, 2;
-4, 10, -6, -2, 2;
6, -16, 10, 4, -6, 8, -10, 4;
-10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2;
12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6;

Mathematica

U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, n*(n - 1)/2 - m]*(-1)^n - U[n - 1, m], 0]
U[1, m_] := U[1, m] = If[m == 0, 1, 0]
Q[n_, m_] := U[n, m - 2] - 2*U[n, m - 1] + U[n, m]
nu[n_]:=(n-1)*n/2+2-n
a[n_, m_] := a[n, m] = If[(m < 0) || (nu[n] < m), 0, a[n - 1, m - n + 1] - a[n - 1, m] - a[n - 1, nu[n - 1]]*Q[n - 1, m]]
a[1, m_] := a[1, m] = If[m == 1, 1, 0]
Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 20}]
Table[a[n, nu[n]], {n, 1, 50}]

Crossrefs

Cf. A000010, A152947, A231599.

Sequence in context: A155682 A191370 A298242 * A288416 A240893 A241108

Adjacent sequences: A282280 A282281 A282282 * A282284 A282285 A282286

Keyword

sign,tabf

Author

Gevorg Hmayakyan, Feb 11 2017

Status

approved