A001191 Digits of positive squares.

1, 4, 9, 1, 6, 2, 5, 3, 6, 4, 9, 6, 4, 8, 1, 1, 0, 0, 1, 2, 1, 1, 4, 4, 1, 6, 9, 1, 9, 6, 2, 2, 5, 2, 5, 6, 2, 8, 9, 3, 2, 4, 3, 6, 1, 4, 0, 0, 4, 4, 1, 4, 8, 4, 5, 2, 9, 5, 7, 6, 6, 2, 5, 6, 7, 6, 7, 2, 9, 7, 8, 4, 8, 4, 1, 9, 0, 0

Offset: 1

Author

Charlie Peck (peck(AT)Alice.Wonderland.Caltech.EDU)

Keywords

• nonn
• base
• easy

Comments

Besicovitch shows that 0.149162536..., this sequence interpreted as a constant, is 10-normal. - Charles R Greathouse IV, Oct 04 2008

The continued fraction of this sequence interpreted as a constant (0.149162536...) displays behavior similar to that of Champernowne's constant, with huge coefficients becoming unbounded: the 47th coefficient has 39 digits, the 103rd coefficient has 178 digits, the 289th coefficient is greater than 10^712, etc. - John M. Campbell, Jun 25 2011

Position of record terms of the continued fraction: 1, 2, 14, 18, 47, 103, 289, 831, 2215, 5801, 14167, 33339, 76595, 174815, 391749, ..., ; Digital length of the record terms: 1, 1, 1, 2, 39, 178, 712, 2637, 9577, 33986, 119198, 413749, 1424714, 4872958, 16572040, ..., . - Robert G. Wilson v, Jul 04 2011

References

• G. Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), 149-166, A K Peters, Natick, MA, 2002.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5500
• A. S. Besicovitch, The asymptotic distribution of the numerals in the decimal representation of the squares of the natural numbers, Mathematische Zeitschrift 39 (1934), pp. 146-156.
• Paul Pollack, Joseph Vandehey, Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)..., arXiv:1405.6266 [math.NT], 2014.
• Paul Pollack, Joseph Vandehey, Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25)..., The American Mathematical Monthly 122.8 (2015): 757-765.

Programs

Mathematica

mx = 30; k = 1; s = 0; While[k < mx+1, s = s (10^IntegerLength[k^2]) + k^2; k++]; IntegerDigits@ s (* Robert G. Wilson v, Jul 04 2011 *)
Flatten[IntegerDigits/@(Range[30]^2)] (* Harvey P. Dale, Aug 14 2014 *)