A117431 String n is at position n in decimal digits of golden ratio (phi).

1, 20, 62, 9956

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..4.

Eric Weisstein's World of Mathematics, The Golden Ratio

Example

1 is a term because the first digit in the golden ratio phi is 1. (phi = 1.6180339887498948482045 ...)

Mathematica

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. *)

Prog

(Python)
from sympy import S
def aupto(nn):
  phistr = str(S.GoldenRatio.n(nn+len(str(nn))+1)).replace(".", "")[:-1]
  for n in range(1, nn+1):
    nstr = str(n)
    if phistr[n-1:n-1+len(nstr)] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021

Crossrefs

Cf. A001622, A057679, A117432.

Sequence in context: A105092 A112144 A007248 * A159504 A182468 A117432

Adjacent sequences: A117428 A117429 A117430 * A117432 A117433 A117434

Keyword

nonn,base,hard,more

Author

Colin Rose, Mar 14 2006

Status

approved