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%I #17 Jan 20 2021 10:47:40
%S 1,20,62,9956
%N String n is at position n in decimal digits of golden ratio (phi).
%C The next such number is greater than 10^7. Not only does number 20 occur at the 20th digit, but it occurs again as the 20th pair of digits (cf. A117432).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">The Golden Ratio</a>
%e 1 is a term because the first digit in the golden ratio phi is 1. (phi = 1.6180339887498948482045 ...)
%t StringFinder[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[(1+Sqrt[5])/2, 10, cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* Example: StringFinder[2] produces all 2-digit members of the sequence. *)
%o (Python)
%o from sympy import S
%o def aupto(nn):
%o phistr = str(S.GoldenRatio.n(nn+len(str(nn))+1)).replace(".", "")[:-1]
%o for n in range(1, nn+1):
%o nstr = str(n)
%o if phistr[n-1:n-1+len(nstr)] == nstr: print(n, end=", ")
%o aupto(10**5) # _Michael S. Branicky_, Jan 20 2021
%Y Cf. A001622, A057679, A117432.
%K nonn,base,hard,more
%O 1,2
%A _Colin Rose_, Mar 14 2006
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