A194679 Triangular array: T(n,k)=[<r^n>+<r^k>], where [ ] = floor, < > = fractional part, and r=3-sqrt(2).

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, 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, 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

Offset

1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for the first 150 rows, flattened

Example

First ten rows:
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 1 1
1 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 1 0 1

Mathematica

r = 3 - Sqrt[2]; z = 15;
p[x_] := FractionalPart[x]; f[x_] := Floor[x];
w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]
Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
  (* A194679 *)
TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
s[n_] := Sum[w[n, k], {k, 1, n}]  (* A194680 *)
Table[s[n], {n, 1, 100}]
h[n_, k_] := f[p[n*r] + p[k*r]]
Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
  (* A194681 *)
TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
t[n_] := Sum[h[n, k], {k, 1, n}]
Table[t[n], {n, 1, 100}]   (* A194682 *)

Prog

(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

Crossrefs

Cf. A194679.

Sequence in context: A174856 A175608 A285467 * A111940 A129572 A070950

Adjacent sequences: A194676 A194677 A194678 * A194680 A194681 A194682

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 01 2011

Status

approved