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

Author

Elliott Line, Mar 25 2019

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• Daniel Suteu, Visual representation of the first 3000 rows

Crossrefs

Cf. A000045, A307069.

Programs

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 *)

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