OFFSET
0,2
COMMENTS
Regarded as binary numbers and converted to decimal, these become 1,5,21,85,... the partial sums of 4^n (see A002450).
Partial sums of 100^n.
Odd terms of A056830. - Alexandre Wajnberg, May 31 2005
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
Previous comment is the answer to the problem A1 proposed during the 50th Putnam Competition in 1989 (link). - Bernard Schott, Mar 24 2023
REFERENCES
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
Robert Price, Table of n, a(n) for n = 0..999
Kiran S. Kedlaya, The 50th William Lowell Putnam Mathematical Competition, Problem A1, Dec 02 1989.
J. V. Leyendekkers and A.G. Shannon, Modular Rings and the Integer 3, Notes on Number Theory & Discrete Mathematics, 17 (2011), 47-51.
Robert Price, Comments on A094028 concerning Elementary Cellular Automata, Feb 21 2016
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (101,-100).
FORMULA
G.f.: 1/((1-x)*(1-100*x)).
a(n) = 1+100*(100^n-1)/99. - N. J. A. Sloane, Apr 20 2008
a(n) = 100^(n+1)/99 - 1/99.
a(n) = A094027(2n+1).
a(n) = 100*a(n-1) + 1, a(0) = 1. - Philippe Deléham, Feb 22 2014
a(n) = 101*a(n-1)-100*a(n-2) for n>1. - Wesley Ivan Hurt, Oct 17 2015
a(n) = (100^(n+1) - 1)/99. - Bernard Schott, Apr 15 2021
EXAMPLE
From Omar E. Pol, Dec 13 2008: (Start)
=======================
n ....... a(n)
0 ........ 1
1 ....... 101
2 ...... 10101
3 ..... 1010101
4 .... 101010101
5 ... 10101010101
======================
(End)
MAPLE
MATHEMATICA
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 *)
PROG
(Maxima) A094028(n):=1+100*(100^n-1)/99$
makelist(A094028(n), n, 0, 30); /* Martin Ettl, Nov 06 2012 */
(Magma) [1+100*(100^n-1)/99 : n in [0..15]]; // Wesley Ivan Hurt, Oct 17 2015
(PARI) a(n) = 1+100*(100^n-1)/99 \\ Felix Fröhlich, Oct 17 2015
(PARI) Vec(1/((1-x)*(1-100*x)) + O(x^100)) \\ Altug Alkan, Oct 17 2015
CROSSREFS
KEYWORD
easy,nonn,changed
AUTHOR
Paul Barry, Apr 22 2004
STATUS
approved