A003076 n-th digit after decimal point of square root of n.

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

Offset

0,4

Comments

Regarded as a decimal fraction, 0.01206932015794604621863800... is likely to be an irrational number.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Formula

From M. F. Hasler, Jun 22 2024: (Start)

a(n) = A000196(n*100^n) % 10, where n % 10 = A010879(n) is the final digit of n.

a(n) = 0 for all n in A000290 (but not only those). This explains that the value 0 is slightly more frequent than the other values. (End)

Example

sqrt(0) = 0.0, so a(0) = 0 with the convention that the 0th digit after the decimal point is the digit before the decimal point,
sqrt(1) = 1.0, where the first digit after the decimal point is a(1) = 0,
sqrt(2) = 1.4142135..., where the 2nd decimal digit is a(2) = 1,
sqrt(3) = 1.7320508..., where the 3rd decimal digit is a(3) = 2,
sqrt(4) = 2.0000000..., where the 4th decimal digit is a(4) = 0,
sqrt(5) = 2.2360679..., where the 5th decimal digit is a(5) = 6,
sqrt(6) = 2.4494897..., where the 6th decimal digit is a(6) = 9, etc.
From M. F. Hasler, Jun 22 2024: (Start)
For the frequency of the respective digits among the first 10^k terms, we have:
  k :   0's    1's   2's    3's   4's   5's   6's   7's   8's   9's
----+---------------------------------------------------------------
  1 :     4,     1,    2,     1,    0,    0,    1,    0,    0,    1;
  2 :    22,     6,    9,    11,   12,    7,    8,    8,    9,    8;
  3 :   126,   106,  105,    94,   95,   90,   86,   96,   92,  110;
  4 :  1097,  1026, 1037,  1031,  984,  979, 1000,  956,  922,  968;
  5 : 10320, 10053, 9926, 10122, 9855, 9985, 9934, 9857, 9855, 10093. (End)

Mathematica

Join[{0}, Array[ Function[ n, RealDigits[ N[ Sqrt[ n ], n+6 ] ]// (#[ [ 1, #[ [ 2 ] ]+n ] ])& ], 110 ]]
Table[ Floor[ Mod[10^n*Sqrt[n], 10]], {n, 0, 104}] (* Robert G. Wilson v, Jun 17 2002 *)

Prog

(PARI) A003076(n)=sqrtint(n*100^n)%10;
apply(A003076, [0..99]) \\ M. F. Hasler, Jun 22 2024

Crossrefs

Cf. A071901.

Sequence in context: A358167 A348189 A372911 * A175478 A011123 A087464

Adjacent sequences: A003073 A003074 A003075 * A003077 A003078 A003079

Keyword

nonn,base,easy

Author

N. J. A. Sloane

Extensions

Extension and program from Olivier Gérard, Oct 15 1997

Status

approved