A216638 First appearance of the Fibonacci numbers in the decimals of Pi.

1, 1, 6, 9, 4, 11, 110, 93, 86, 130, 11, 1638, 229, 3056, 268, 1510, 10118, 11477, 727, 17711, 83295, 59861, 22334, 19659, 301848, 977089, 59943, 414086, 536681, 649382, 2729036, 68232754, 17793212, 33986473, 695781, 135830965, 117951651, 36978613, 170243036, 366567058

Offset: 1

Author

Vicente Izquierdo Gomez, Sep 11 2012

Keywords

• nonn
• base

Examples

			Fibonacci(4) is 3, 3 appears for the first time in decimals of Pi in position 9, so a(4) = 9.
		

Links

• Peter Trüb, 22.4 trillion digits of pi.

Crossrefs

Cf. A000045, A000796, A014777.

Programs

Mathematica

(* Determine the decimal digits of Pi following the decimal point. *)
decimalPiDigits[n_] := First@RealDigits[Pi, 10, n, -1];
(* Find the position of first occurrence of 'sublist' in 'list', or Indeterminate if it doesn't occur. *)
firstPosition[sublist_, list_] :=
  With[{p = SequencePosition[list, sublist]},
   If[Length[p] == 0, Indeterminate, First@First@p]];
(* Find the first occurrence of the given digits in the decimal digits of Pi by calculating ever more digits of Pi, as needed. *)
findDigitSequenceInDecimalPiDigits[seq_] :=
  First@NestWhile[
    With[
      {
       numdigits = Max[1, 2*Last[#]] (*
       How many digits will we calculate in this iteration? *)
       },
      {firstPosition[seq, decimalPiDigits[numdigits]], numdigits}
      ] &,
    {Indeterminate, 0},
    Not@*IntegerQ@*First
    ];
(* Find the first 30 entries. *)
Table[findDigitSequenceInDecimalPiDigits[
  IntegerDigits@Fibonacci[n]], {n, 1, 30}]
(* Sidney Cadot, Feb 25 2023 *)

Formula

a(n) = A014777(A000045(n)). - Pontus von Brömssen, Aug 31 2024

Extensions

a(31)-a(40) from Pontus von Brömssen, Aug 31 2024