A282283 - OEIS

%I #10 Feb 18 2017 22:35:48

%S 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,

%T -10,10,-2,8,-10,0,2,12,-34,22,10,-12,12,-22,30,-30,6,10,-10,8,0,6,

%U -14,6,-18,52,-34,-16,18,-18,34,-36,20,10,-6,-2,4,-28,18,8

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

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

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

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

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

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

%F 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)

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

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

%e The first few rows are:

%e 0, 1;

%e -1, 1;

%e 2, -4, 2;

%e -4, 10, -6, -2, 2;

%e 6, -16, 10, 4, -6, 8, -10, 4;

%e -10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2;

%e 12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6;

%t U[n_, m_] := U[n, m] = If[n > 1, U[n - 1, n*(n - 1)/2 - m]*(-1)^n - U[n - 1, m], 0]

%t U[1, m_] := U[1, m] = If[m == 0, 1, 0]

%t Q[n_, m_] := U[n, m - 2] - 2*U[n, m - 1] + U[n, m]

%t nu[n_]:=(n-1)*n/2+2-n

%t 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]]

%t a[1, m_] := a[1, m] = If[m == 1, 1, 0]

%t Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 20}]

%t Table[a[n, nu[n]], {n, 1, 50}]

%Y Cf. A000010, A152947, A231599.

%K sign,tabf

%O 0,5

%A _Gevorg Hmayakyan_, Feb 11 2017