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A094028 Expansion of 1/((1-x)*(1-100*x)). 48
1, 101, 10101, 1010101, 101010101, 10101010101, 1010101010101, 101010101010101, 10101010101010101, 1010101010101010101, 101010101010101010101, 10101010101010101010101 (list; graph; refs; listen; history; text; internal format)
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

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

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).

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

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: A262861 A164367 A263244 * A144564 A261965 A255892

Adjacent sequences:  A094025 A094026 A094027 * A094029 A094030 A094031

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 22 2004

STATUS

approved

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Last modified October 15 13:01 EDT 2018. Contains 316236 sequences. (Running on oeis4.)