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

%I A216638 #20 Aug 31 2024 08:30:42
%S A216638 1,1,6,9,4,11,110,93,86,130,11,1638,229,3056,268,1510,10118,11477,727,
%T A216638 17711,83295,59861,22334,19659,301848,977089,59943,414086,536681,
%U A216638 649382,2729036,68232754,17793212,33986473,695781,135830965,117951651,36978613,170243036,366567058
%N A216638 First appearance of the Fibonacci numbers in the decimals of Pi.
%H A216638 Peter Trüb, <a href="https://pi2e.ch/blog/2017/03/10/pi-digits-download/">22.4 trillion digits of pi</a>.
%F A216638 a(n) = A014777(A000045(n)). - _Pontus von Brömssen_, Aug 31 2024
%e A216638 Fibonacci(4) is 3, 3 appears for the first time in decimals of Pi in position 9, so a(4) = 9.
%t A216638 (* Determine the decimal digits of Pi following the decimal point. *)
%t A216638 decimalPiDigits[n_] := First@RealDigits[Pi, 10, n, -1];
%t A216638 (* Find the position of first occurrence of 'sublist' in 'list', or Indeterminate if it doesn't occur. *)
%t A216638 firstPosition[sublist_, list_] :=
%t A216638   With[{p = SequencePosition[list, sublist]},
%t A216638    If[Length[p] == 0, Indeterminate, First@First@p]];
%t A216638 (* Find the first occurrence of the given digits in the decimal digits of Pi by calculating ever more digits of Pi, as needed. *)
%t A216638 findDigitSequenceInDecimalPiDigits[seq_] :=
%t A216638   First@NestWhile[
%t A216638     With[
%t A216638       {
%t A216638        numdigits = Max[1, 2*Last[#]] (*
%t A216638        How many digits will we calculate in this iteration? *)
%t A216638        },
%t A216638       {firstPosition[seq, decimalPiDigits[numdigits]], numdigits}
%t A216638       ] &,
%t A216638     {Indeterminate, 0},
%t A216638     Not@*IntegerQ@*First
%t A216638     ];
%t A216638 (* Find the first 30 entries. *)
%t A216638 Table[findDigitSequenceInDecimalPiDigits[
%t A216638   IntegerDigits@Fibonacci[n]], {n, 1, 30}]
%t A216638 (* _Sidney Cadot_, Feb 25 2023 *)
%Y A216638 Cf. A000045, A000796, A014777.
%K A216638 nonn,base
%O A216638 1,3
%A A216638 _Vicente Izquierdo Gomez_, Sep 11 2012
%E A216638 a(31)-a(40) from _Pontus von Brömssen_, Aug 31 2024