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A307116 A special version of Pascal's triangle where only Fibonacci numbers are permitted. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 23 19:45 EST 2020. Contains 331175 sequences. (Running on oeis4.)