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A194680 Number of k in [1,n] for which + > 1, where < > = fractional part, and r=3-sqrt(2).

%I A194680 #9 Feb 09 2018 03:21:02
%S A194680 1,2,3,1,1,5,2,8,4,6,5,11,2,12,8,5,3,12,8,5,13,5,23,18,16,6,6,18,21,
%T A194680 27,7,24,12,4,23,7,26,35,31,20,41,29,9,5,19,9,47,37,5,24,24,37,50,46,
%U A194680 22,17,9,43,53,7,55,16,44,42,4,46,13,48,46,50,63,17,57,43,38,9
%N A194680 Number of k in [1,n] for which <r^n>+<r^k> > 1, where < > = fractional part, and r=3-sqrt(2).
%H A194680 G. C. Greubel, <a href="/A194680/b194680.txt">Table of n, a(n) for n = 1..5000</a>
%t A194680 r = 3 - Sqrt[2]; z = 15;
%t A194680 p[x_] := FractionalPart[x]; f[x_] := Floor[x];
%t A194680 w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]
%t A194680 Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t A194680   (* A194679 *)
%t A194680 TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
%t A194680 s[n_] := Sum[w[n, k], {k, 1, n}]  (* A194680 *)
%t A194680 Table[s[n], {n, 1, 100}]
%t A194680 h[n_, k_] := f[p[n*r] + p[k*r]]
%t A194680 Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
%t A194680   (* A194681 *)
%t A194680 TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
%t A194680 t[n_] := Sum[h[n, k], {k, 1, n}]
%t A194680 Table[t[n], {n, 1, 100}]   (* A194682 *)
%o A194680 (PARI) for(n=1, 30, print1(round(sum(k=1, n, 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 A194680 Cf. A194679.
%K A194680 nonn
%O A194680 1,2
%A A194680 _Clark Kimberling_, Sep 01 2011