|
%I #9 Feb 12 2013 16:00:46
%S 1,1,1,1,0,1,1,1,1,1,1,-1,0,-1,1,1,3,2,2,3,1,1,-3,0,-2,0,-3,1,1,3,0,2,
%T 2,0,3,1,1,-1,2,-2,2,-2,2,-1,1,1,3,2,4,2,2,4,2,3,1,1,-5,-2,-4,0,-6,0,
%U -4,-2,-5,1
%N Triangle T(n,m) = - phi(n+1) + phi(m+1) + phi(n-m+1), 0<=m<=n, where phi = A000010 is Euler's totient.
%C Row sums are 1, 2, 2, 4, 0, 12, -6, 12, 2, 24, -26,...
%F T(n,m) = T(n,n-m).
%F Sum_{m=0..n} T(n,m) = 2*A002088(n+1)-A002618(n+1). - _R. J. Mathar_, Feb 12 2013
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, 1, 1, 1;
%e 1, -1, 0, -1, 1;
%e 1, 3, 2, 2, 3, 1;
%e 1, -3, 0, -2, 0, -3, 1;
%e 1, 3, 0, 2, 2, 0, 3, 1;
%e 1, -1, 2, -2, 2, -2, 2, -1, 1;
%e 1, 3, 2, 4, 2, 2, 4, 2, 3, 1;
%e 1, -5, -2, -4, 0, -6, 0, -4, -2, -5, 1;
%p A178086 := proc(n,m)
%p -numtheory[phi](n+1)+numtheory[phi](m+1)+numtheory[phi](n-m+1)
%p end proc;
%p seq(seq(A178086(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Feb 12 2013
%t T[n_, m_, q_] := 1 - EulerPhi[n + q] + (EulerPhi[m + q] + EulerPhi[n - m + q]) - EulerPhi[q];
%t Table[Flatten[Table[Table[T[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
%o (PARI) T(n,m)=-eulerphi(n+1)+eulerphi(m+1)+eulerphi(n-m+1) \\ _Charles R Greathouse IV_, Feb 12 2013
%K sign,tabl,easy
%O 0,17
%A _Roger L. Bagula_, May 19 2010
|
