A307069 Given a special version of Pascal's triangle where only Fibonacci numbers are permitted, a(n) is the row number in which the n-th Fibonacci number first appears.

0, 0, 2, 3, 9, 50, 51, 70, 71, 133, 134, 135, 136, 2543, 2544

Offset: 1

Author

Elliott Line, Mar 22 2019

Keywords

• nonn
• more

Comments

Consider a version of Pascal's Triangle: a triangular array with a single 1 on row 0, with numbers below equal to the sum of the two numbers above it if and only if that sum appears in the Fibonacci sequence A000045. If the sum does not appear in A000045, a 1 is put in its place.

So the first few rows would be 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

...

a(n) is the row number in which the n-th Fibonacci number first appears in this triangular array.

a(16) > 2.2*10^5. - David A. Corneth, Mar 25 2019

a(16) > 3.2*10^6. - Daniel Suteu, Mar 26 2019

a(16) > 1.5*10^7. - Bert Dobbelaere, Apr 02 2019

Links

• David A. Corneth, PARI program

Crossrefs

Cf. A000045, A307116 (the special Pascal's triangle).

Programs

Mathematica

Block[{s = Array[Fibonacci, 20], t}, t = Nest[Append[#1, (PadLeft[#1[[-1]], #2] + PadRight[#1[[-1]], #2]) /. k_Integer /; FreeQ[s, k] -> 1] & @@ {#, Length@ # + 1} &, {{1}}, 10^4]; -1 + TakeWhile[Map[FirstPosition[t, #][[1]] &, s], IntegerQ]] (* Michael De Vlieger, Mar 24 2019 *)

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
lista(nn) = {print1(0, ", ", 0, ", "); v = [1,1]; nextf = 3; for (n=2, nn, w = vector(n+1); w[1] = v[1]; for (j=2, n, w[j] = v[j-1]+ v[j]; if (!isfib(w[j]), w[j] = 1)); w[n+1] = v[n]; sw = vecsort(w,,8); if (vecsearch(sw, fibonacci(nextf)), print1(n, ", "); nextf++); v = w;);} \\ Michel Marcus, Mar 22 2019

PARI

See Corneth link \\ David A. Corneth, Mar 25 2019

Extensions

a(14)-a(15) from Michel Marcus, Mar 22 2019