A094028 - OEIS

%I #63 Mar 25 2023 08:20:09

%S 1,101,10101,1010101,101010101,10101010101,1010101010101,

%T 101010101010101,10101010101010101,1010101010101010101,

%U 101010101010101010101,10101010101010101010101,1010101010101010101010101,101010101010101010101010101,10101010101010101010101010101

%N Expansion of 1/((1-x)*(1-100*x)).

%C Regarded as binary numbers and converted to decimal, these become 1,5,21,85,... the partial sums of 4^n (see A002450).

%C Partial sums of 100^n.

%C Odd terms of A056830. - _Alexandre Wajnberg_, May 31 2005

%C 101 is the only term that is prime, since (100^k-1)/99 = (10^k+1)/11 * (10^k-1)/9. When k is odd and not 1, (10^k+1)/11 is an integer > 1 and thus (100^k-1)/99 is nonprime. When k is even and greater than 2, (100^k-1)/99 has the prime factor 101 and is nonprime. - _Felix Fröhlich_, Oct 17 2015

%C Previous comment is the answer to the problem A1 proposed during the 50th Putnam Competition in 1989 (link). - _Bernard Schott_, Mar 24 2023

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A094028/b094028.txt">Table of n, a(n) for n = 0..999</a>

%H Kiran S. Kedlaya, <a href="http://kskedlaya.org/putnam-archive/1989.pdf">The 50th William Lowell Putnam Mathematical Competition</a>, Problem A1, Dec 02 1989.

%H J. V. Leyendekkers and A.G. Shannon, <a href="http://www.nntdm.net/papers/nntdm-17/NNTDM-17-2-47-51.pdf">Modular Rings and the Integer 3</a>, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51.

%H Robert Price, <a href="/A094028/a094028_1.txt">Comments on A094028 concerning Elementary Cellular Automata</a>, Feb 21 2016

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (101,-100).

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical Competitions</a>.

%F G.f.: 1/((1-x)*(1-100*x)).

%F a(n) = 1+100*(100^n-1)/99. - _N. J. A. Sloane_, Apr 20 2008

%F a(n) = 100^(n+1)/99 - 1/99.

%F a(n) = A094027(2n+1).

%F a(n) = 100*a(n-1) + 1, a(0) = 1. - _Philippe Deléham_, Feb 22 2014

%F a(n) = 101*a(n-1)-100*a(n-2) for n>1. - _Wesley Ivan Hurt_, Oct 17 2015

%F a(n) = (100^(n+1) - 1)/99. - _Bernard Schott_, Apr 15 2021

%e From _Omar E. Pol_, Dec 13 2008: (Start)

%e =======================

%e n ....... a(n)

%e 0 ........ 1

%e 1 ....... 101

%e 2 ...... 10101

%e 3 ..... 1010101

%e 4 .... 101010101

%e 5 ... 10101010101

%e ======================

%e (End)

%p A094028:=n->1+100*(100^n-1)/99: seq(A094028(n), n=0..15); # _Wesley Ivan Hurt_, Oct 17 2015

%t CoefficientList[Series[1/((1-x)(1-100x)),{x,0,20}],x] (* or *) Table[ FromDigits[ PadRight[{},2n-1,{1,0}]],{n,20}] (* or *) LinearRecurrence[ {101,-100},{1,101},20] (* or *) NestList[100#+1&,1,20] (* _Harvey P. Dale_, Apr 27 2015 *)

%o (Maxima) A094028(n):=1+100*(100^n-1)/99$

%o makelist(A094028(n),n,0,30); /* _Martin Ettl_, Nov 06 2012 */

%o (Magma) [1+100*(100^n-1)/99 : n in [0..15]]; // _Wesley Ivan Hurt_, Oct 17 2015

%o (PARI) a(n) = 1+100*(100^n-1)/99 \\ _Felix Fröhlich_, Oct 17 2015

%o (PARI) Vec(1/((1-x)*(1-100*x)) + O(x^100)) \\ _Altug Alkan_, Oct 17 2015

%Y Bisection of A147759. [_Omar E. Pol_, Nov 13 2008]

%Y Cf. A002450, A056830, A094027.

%Y Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 22 2004