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

1,2

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)

Links

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

Formula

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

Example

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)

Mathematica

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

Prog

(Magma) [-(199/9801)-(1/99)*n+(10000/9801)*100^n: n in [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

Crossrefs

Cf. A007908, A014824, A262581, A262582.

Sequence in context: A284448 A274252 A303993 * A097725 A353142 A129751

Adjacent sequences: A030509 A030510 A030511 * A030513 A030514 A030515

Keyword

nonn,fini,full,base

Author

Ralf Stephan

Extensions

Edited by Charles R Greathouse IV, Apr 28 2010

Limits corrected by Georg Fischer, Mar 10 2020

Status

approved