A030512 Concatenation of first n 2-digit positive integers including leading zeros.

1, 102, 10203, 1020304, 102030405, 10203040506, 1020304050607, 102030405060708, 10203040506070809, 1020304050607080910, 102030405060708091011, 10203040506070809101112, 1020304050607080910111213, 102030405060708091011121314

Offset: 1

Author

Ralf Stephan

Keywords

• nonn
• fini
• full
• base

Comments

From Peter Bala, Sep 14 2015: (Start)

Empirically, we observe that the square roots of these numbers and their reciprocals have some interesting properties, as follows (examples are given below).

The decimal expansion of sqrt(a(n)) begins with strings of repeated digits (that gradually shorten in length until they disappear) alternating with strings of apparently random digits.

The decimal expansion of 1/sqrt(a(n)) has long strings of 0's (gradually shortening in length until they disappear) interspersed with blocks of digits. If we read these blocks of digits as ordinary integers and factorize them, we find the numbers are related in a surprising manner. Cf. A014824.

(End)

Examples

			From _Peter Bala_, Sep 14 2015: (Start)
Decimal expansions with repeating strings of digits in parentheses for clarity:
sqrt(a(50)) = 1.(0101...0101)0075(5050...5050)4728503 (7878...7878)7065734690(6565...6565)63090366531526199 (4949...4949)40423435587935014204(5454...5454) 511096186531728108723958(33...33)197004273464583079020182291 (66...66)107291492892700779438018798828124(99...99) 7645962810367893557912773556179470486(11...11) 010064064746152... * 10^49.
1/sqrt(a(10))  = 9.9(0...0)53955(0...0)441082125(0..0)4... * 10^(-10). The long strings of zeros gradually shorten in length until they disappear and are interspersed with five blocks of digits [99, 53955, 441082125, 400649596875, 38211955301953125] = [3^2*11, 3^2*5*11*109, 3^3*5^3*11*109^2, 3^2*5^5*11*109^3, 3^2*5^8*7*11*109^4].
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..99 [Truncated to 99 terms by _Georg Fischer_, Mar 10 2020]

Crossrefs

Cf. A007908, A014824, A262581, A262582.

Programs

Magma

[-(199/9801)-(1/99)*n+(10000/9801)*100^n: n in [0..98]]; // Vincenzo Librandi, May 17 2013

Mathematica

Table[-(199/9801)-(1/99) n + (10000/9801) 100^n, {n, 0, 98}] (* Vincenzo Librandi, May 17 2013 *)

PARI

a(n) = -(199/9801) - (1/99)*(n-1) + (10000/9801)*100^(n-1);
vector(20, n, a(n)) \\ Altug Alkan, Oct 01 2015

Formula

a(n+1) = 100*a(n) + n + 1 for n<100.

Extensions

Edited by Charles R Greathouse IV, Apr 28 2010

Limits corrected by Georg Fischer, Mar 10 2020