A094028 Expansion of 1/((1-x)*(1-100*x)).

1, 101, 10101, 1010101, 101010101, 10101010101, 1010101010101, 101010101010101, 10101010101010101, 1010101010101010101, 101010101010101010101, 10101010101010101010101, 1010101010101010101010101, 101010101010101010101010101, 10101010101010101010101010101

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 to Elementary Cellular Automata

Index entries for sequences related to cellular automata

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Index to sequences related to Olympiads and other Mathematical Competitions.

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

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

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

Bisection of A147759. [Omar E. Pol, Nov 13 2008]

Cf. A002450, A056830, A094027.

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

Sequence in context: A164367 A263244 A368417 * A144564 A261965 A255892

Adjacent sequences: A094025 A094026 A094027 * A094029 A094030 A094031

Keyword

easy,nonn

Author

Paul Barry, Apr 22 2004

Status

approved