A209355 Sequence with each term appearing in runs of every length infinitely often.

0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset

1

Comments

This construction is in stages for sequence range 0, 1:

Stage 1: 0, 1

Stage 2: 0, 1, 0, 0, 1, 1

Stage 3: 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, etc.,

where each stage is the previous stage (if any) followed by stage-number of zeros then stage-number of ones. The sequence is all stages read sequentially as rows. See PARI program for comments about stage lengths and cumulative lengths.

For each positive integer n each term appears in runs of length exactly n infinitely often. This sequence's construction process, which can clearly be generalized for any (finite) number of distinct terms, has A079813 as the limit of its n-th stage as n --> infinity.

Links

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Example

Terms a(15) through a(17) are the first run (of infinitely many such runs) of exactly three zeros.

Prog

(PARI)
/* range = vector of r >= 2 integers in desired order. */
/*(r = 0, 1 allowed but [], constant seq., resp., result)*/
/* stages_wanted >= 1 (or empty vector returned).      */
/* Stage (Row) sizes for r = 2 integers in range are   */
/* 2, 6, 12, 20, 30, 42, ... (A002378(stage) for each stage), */
/* producing vector v's A007290(stages_wanted+2) terms.*/
/*                                                     */
{a(range, stages_wanted) =
local(v = [], next_stage = [], stage = 0, k, m);
while(stage<stages_wanted,
  stage++;
  for(k=1, matsize(range)[2],
   next_stage=concat(next_stage, vector(stage, m, range[k]))
  ); v = concat(v, next_stage);
); return(v)}
/* A209355 = a([0, 1], 6); /* 112= 2+6+12+20+30+42 terms*/
A209355 = a([0, 1], 31);   /* 10912 = A007290(33)terms  */
for(n=1, 10000, write("b209355.txt", n, " ", A209355[n]))

Crossrefs

Cf. A079813, A002378, A007290.

Sequence in context: A173922 A213676 A309752 * A141743 A268384 A358670

Adjacent sequences: A209352 A209353 A209354 * A209356 A209357 A209358

Keyword

nonn,easy

Author

Rick L. Shepherd, Mar 06 2012

Status

approved