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A194675 Triangular array: T(n,k)=[<e^n>+<e^k>], where [ ] = floor, < > =  fractional part. 5

%I

%S 1,1,0,0,0,0,1,0,0,1,1,0,0,1,0,1,0,0,1,0,0,1,1,0,1,1,1,1,1,1,1,1,1,1,

%T 1,1,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,

%U 0,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,0,1

%N Triangular array: T(n,k)=[<e^n>+<e^k>], where [ ] = floor, < > = fractional part.

%C n-th row sum gives number of k in [0,1] for which <e^n>+<e^k> > 1; see A194676.

%e First ten rows:

%e 1

%e 1 0

%e 0 0 0

%e 1 0 0 1

%e 1 0 0 1 0

%e 1 0 0 1 0 0

%e 1 1 0 1 1 1 1

%e 1 1 1 1 1 1 1 1

%e 0 0 0 0 0 0 0 1 0

%e 1 0 0 1 0 0 1 1 0 0

%t r = E; z = 15;

%t p[x_] := FractionalPart[x]; f[x_] := Floor[x];

%t w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]

%t Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]] (* A194675 *)

%t TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]

%t s[n_] := Sum[w[n, k], {k, 1, n}] (* A194676 *)

%t Table[s[n], {n, 1, 100}]

%t h[n_, k_] := f[p[n*r] + p[k*r]]

%t Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]

%t (* A194677 *)

%t TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]

%t t[n_] := Sum[h[n, k], {k, 1, n}]

%t Table[t[n], {n, 1, 100}] (* A194678 *)

%Y Cf. A194676.

%K nonn,tabl

%O 1

%A _Clark Kimberling_, Sep 01 2011

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Last modified October 20 07:33 EDT 2019. Contains 328252 sequences. (Running on oeis4.)