A194679 - OEIS

%I #9 Feb 09 2018 03:20:58

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

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

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

%N Triangular array: T(n,k)=[<r^n>+<r^k>], where [ ] = floor, < > =  fractional part, and r=3-sqrt(2).

%C n-th row sum gives number of k in [0,1] for which <r^n>+<r^k> > 1; see A194680.

%H G. C. Greubel, <a href="/A194679/b194679.txt">Table of n, a(n) for the first 150 rows, flattened</a>

%e First ten rows:

%e 1

%e 1 1

%e 1 1 1

%e 0 0 1 0

%e 0 0 1 0 0

%e 1 1 1 1 0 1

%e 0 0 1 0 0 1 0

%e 1 1 1 1 1 1 1 1

%e 1 0 1 0 0 1 0 1 0

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

%t r = 3 - Sqrt[2]; 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}]]

%t   (* A194679 *)

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

%t s[n_] := Sum[w[n, k], {k, 1, n}]  (* A194680 *)

%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   (* A194681 *)

%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}]   (* A194682 *)

%o (PARI) for(n=1, 20, for(k=1, n, print1(round(frac((3-sqrt(2))^n) + frac((3-sqrt(2))^k) - frac((3-sqrt(2))^n + (3-sqrt(2))^k)), ", "))) \\ _G. C. Greubel_, Feb 08 2018

%Y Cf. A194679.

%K nonn,tabl

%O 1

%A _Clark Kimberling_, Sep 01 2011