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
KEYWORD
nonn,fini,full,base
AUTHOR
EXTENSIONS
Edited by Charles R Greathouse IV, Apr 28 2010
Limits corrected by Georg Fischer, Mar 10 2020
STATUS
approved