cp's OEIS Frontend

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

%I A282283 #10 Feb 18 2017 22:35:48
%S A282283 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 A282283 -10,10,-2,8,-10,0,2,12,-34,22,10,-12,12,-22,30,-30,6,10,-10,8,0,6,
%U A282283 -14,6,-18,52,-34,-16,18,-18,34,-36,20,10,-6,-2,4,-28,18,8
%N A282283 Recursive 2-parameter sequence allowing calculation of the Euler Totient function.
%C A282283 The a(n,m) forms a table where each row has (n*(n-3)+4)/2 = A152947(n) elements.
%C A282283 The index of the first row is n=1 and the index of the first column is m=0.
%C A282283 The right diagonal a(n, A152947(n)) = A000010(n), Euler Totient function.
%F A282283 nu(n) = (n*(n-3)+4)/2
%F A282283 Q(n,m) = 2*A231599(n,m-1)-A231599(n,m-2)-A231599(n,m)
%F A282283 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 A282283 a(1,m)=1 if m=1 and 0 otherwise.
%F A282283 a(n,nu(n))= A000010(n)
%e A282283 The first few rows are:
%e A282283 0, 1;
%e A282283 -1, 1;
%e A282283 2, -4, 2;
%e A282283 -4, 10, -6, -2, 2;
%e A282283 6, -16, 10, 4, -6, 8, -10, 4;
%e A282283 -10, 28, -18, -8, 10, -10, 10, -2, 8, -10, 0, 2;
%e A282283 12, -34, 22, 10, -12, 12, -22, 30, -30, 6, 10, -10, 8, 0, 6, -14, 6;
%t A282283 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 A282283 U[1, m_] := U[1, m] = If[m == 0, 1, 0]
%t A282283 Q[n_, m_] := U[n, m - 2] - 2*U[n, m - 1] + U[n, m]
%t A282283 nu[n_]:=(n-1)*n/2+2-n
%t A282283 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 A282283 a[1, m_] := a[1, m] = If[m == 1, 1, 0]
%t A282283 Table[Table[a[n, m], {m, 0, nu[n]}], {n, 1, 20}]
%t A282283 Table[a[n, nu[n]], {n, 1, 50}]
%Y A282283 Cf. A000010, A152947, A231599.
%K A282283 sign,tabf
%O A282283 0,5
%A A282283 _Gevorg Hmayakyan_, Feb 11 2017