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

%I A030512 #45 Feb 18 2024 02:04:40
%S A030512 1,102,10203,1020304,102030405,10203040506,1020304050607,
%T A030512 102030405060708,10203040506070809,1020304050607080910,
%U A030512 102030405060708091011,10203040506070809101112,1020304050607080910111213,102030405060708091011121314
%N A030512 Concatenation of first n 2-digit positive integers including leading zeros.
%C A030512 From _Peter Bala_, Sep 14 2015: (Start)
%C A030512 Empirically, we observe that the square roots of these numbers and their reciprocals have some interesting properties, as follows (examples are given below).
%C A030512 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.
%C A030512 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.
%C A030512 (End)
%H A030512 Vincenzo Librandi, <a href="/A030512/b030512.txt">Table of n, a(n) for n = 1..99</a> [Truncated to 99 terms by _Georg Fischer_, Mar 10 2020]
%F A030512 a(n+1) = 100*a(n) + n + 1 for n<100.
%e A030512 From _Peter Bala_, Sep 14 2015: (Start)
%e A030512 Decimal expansions with repeating strings of digits in parentheses for clarity:
%e A030512 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.
%e A030512 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].
%e A030512 (End)
%t A030512 Table[-(199/9801)-(1/99) n + (10000/9801) 100^n, {n, 0, 98}] (* _Vincenzo Librandi_, May 17 2013 *)
%o A030512 (Magma) [-(199/9801)-(1/99)*n+(10000/9801)*100^n: n in [0..98]]; // _Vincenzo Librandi_, May 17 2013
%o A030512 (PARI) a(n) = -(199/9801) - (1/99)*(n-1) + (10000/9801)*100^(n-1);
%o A030512 vector(20, n, a(n)) \\ _Altug Alkan_, Oct 01 2015
%Y A030512 Cf. A007908, A014824, A262581, A262582.
%K A030512 nonn,fini,full,base
%O A030512 1,2
%A A030512 _Ralf Stephan_
%E A030512 Edited by _Charles R Greathouse IV_, Apr 28 2010
%E A030512 Limits corrected by _Georg Fischer_, Mar 10 2020