A307116 A special version of Pascal's triangle where only Fibonacci numbers are permitted.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 1, 3, 2, 2, 1, 1, 3, 1, 5, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 3, 1, 1, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1

Offset

0,5

Comments

If the sum of the two numbers above in the triangular array is not a Fibonacci number (A000045), then a 1 is put in its place.

A307069(k) is the row number of the first instance of the k-th Fibonacci number.

Links

Seiichi Manyama, Rows n = 0..139, flattened

Daniel Suteu, Visual representation of the first 3000 rows

Example

The first few rows are as follows:
  row 0:            1
  row 1:           1 1
  row 2:          1 2 1
  row 3:         1 3 3 1
  row 4:        1 1 1 1 1
  row 5:       1 2 2 2 2 1
  row 6:      1 3 1 1 1 3 1
  row 7:     1 1 1 2 2 1 1 1
  row 8:    1 2 2 3 1 3 2 2 1
  row 9:   1 3 1 5 1 1 5 1 3 1

Mathematica

With[{s = Array[Fibonacci, 12]}, Nest[Append[#, Join[{1}, Map[Total[#] /. k_ /; FreeQ[s, k] -> 1 &, Partition[#[[-1]], 2, 1]], {1}]] &, {{1}}, 12]] // Flatten (* Michael De Vlieger, Mar 28 2019 *)

Prog

(PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
rows(nn) = {v = [1]; print(v); if (nn == 1, return); v = [1, 1]; print(v); if (nn == 2, return); for (n=3, nn, w = vector(n); w[1] = v[1]; for (j=2, n-1, w[j] = v[j-1]+ v[j]; if (!isfib(w[j]), w[j] = 1); ); w[n] = v[n-1]; print(w); v = w; ); } \\ Michel Marcus, Mar 28 2019

Crossrefs

Cf. A000045, A307069.

Sequence in context: A180180 A034931 A248473 * A212626 A090402 A026082

Adjacent sequences: A307113 A307114 A307115 * A307117 A307118 A307119

Keyword

nonn,tabl

Author

Elliott Line, Mar 25 2019

Status

approved