A126999 Zero-one fractional-part array for the golden ratio; a rectangular array T by antidiagonals.

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1

Offset

1,1

Comments

(Row 1) = (Column 1) = A005614 (infinite Fibonacci word).

(Row 2) = (Column 2) = A123740

(Row 3) = (Column 3) = A187944

(Row 4) = (Column 4) = A187950

(Main Diagonal) = A078588

See A187950 for connections to left-shifted sums of the infinite Fibonacci word.

Links

G. C. Greubel, Rows n = 1..100 of antidiagonals, flattened

Formula

T(n,k) = {nr} + {kr} - {nr+kr}, where r=(1+sqrt(5))/2 and { } denotes fractional part;, i.e., {x} = x - floor(x).

T(n,k) = [nr] + [kr] - [nr+kr], where []=floor.

Example

Northwest corner:
  1 0 1 1 0 1 0 1 1
  0 0 1 0 0 0 0 1 0
  1 1 1 1 0 1 1 1 1
  1 0 1 0 0 1 0 1 1
  0 0 0 0 0 0 0 1 0
  1 0 1 1 0 1 1 1 1
T(3,3)=1 because 2{3x}-{6x}=1.
The antidiagonals form a triangle with these first six rows:
  1
  0 0
  1 0 1
  1 1 1 1
  0 0 1 0 0
  1 0 1 1 0 1

Mathematica

r:=(1+Sqrt[5])/2;
T[k_, n_]:=Floor[n*r+k*r]-Floor[n*r]-Floor[k*r]
TableForm[Table[T[n, k], {k, 1, 10}, {n, 1, 10}]]
Table[T[n-k+1, k], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, May 30 2019 *)

Prog

(PARI) r=(1+sqrt(5))/2;
T(n, k) = ((n+k)*r)\1 - (n*r)\1 - (k*r)\1;
for(n=1, 10, for(k=1, n, print1(T(n-k+1, k), ", "))) \\ G. C. Greubel, May 30 2019
(Magma) r:=(1+Sqrt(5))/2; [[Floor((n+1)*r)-Floor((n-k+1)*r)-Floor(k*r): k in [1..n]]: n in [1..10]]; // G. C. Greubel, May 30 2019
(Sage)
def T(n, k): return floor((n+k)*golden_ratio) - floor(n*golden_ratio) - floor(k*golden_ratio)
[[T(n-k+1, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 30 2019

Crossrefs

Cf. A005614, A123740, A078588, A126700, A126701, A188294, A187950.

Sequence in context: A347523 A143221 A325899 * A306862 A363801 A347283

Adjacent sequences: A126996 A126997 A126998 * A127000 A127001 A127002

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 01 2007

Status

approved