A194680 - OEIS

%I #9 Feb 09 2018 03:21:02

%S 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 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 22,17,9,43,53,7,55,16,44,42,4,46,13,48,46,50,63,17,57,43,38,9

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

%H G. C. Greubel, <a href="/A194680/b194680.txt">Table of n, a(n) for n = 1..5000</a>

%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, 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 Cf. A194679.

%K nonn

%O 1,2

%A _Clark Kimberling_, Sep 01 2011