A187188 Parse the infinite string 0123456789012345678901234567890... into distinct phrases 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 01, 23, 45, 67, 89, 012, 34, 56, 78, 90, 12, 345, ...; a(n) = length of n-th phrase.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 5, 6, 5, 5, 6, 5, 5, 6, 5, 5, 6, 5, 5, 6, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 12, 12, 12, 12, 12, 13, 12, 12, 12, 12

Offset

1,11

Comments

See A187180-A187187 for further details.

Answers a question raised by Sergio Verdu (personal communication, Mar 05 2011).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Formula

After the initial block of 10 1's, the sequence is quasi-periodic with period 100, increasing by 10 after each block. In more detail:

a(n) = 1 for 1 <= n <= 10.

For n >= 10, write n = 11 + 100i + j with i >= 0, 0 <= j <= 99.

Then for 0 <= j <= 79, a(n) = 10i + f(j),

where f(0) ... f(79) is the following 80-term sequence:

[2 2 2 2 2 3 2 2 2 2 2 3

3 3 3 3 3 3 3 3

4 4 4 4 4 5 4 4 4 4 4 5

6 5 5 6 5 5 6 5 5 6 5 5

6 7

6 6 6 6 6

7 7 7 7 7 7 7 7 7

8 8 8 8 8 9 8 8 8 8 8 9

9 9 9 9 9 9 9 9]

(this has been broken into blocks to make it easier to see),

and for 80 <= j <= 99, a(n) = 10i+10 if j is even, a(n) = 10i+11 if j is odd.

Examples:

n=120 = 11 + 100*1 + 9, i=1, j=9, a(120)=10+f(9) = 10+2 = 12

n=292 = 11 + 100*2 + 81, i=2, j=81. a(292)=20+11=31

Example

The sequence begins
1   1   1   1   1   1   1   1   1   1
2   2   2   2   2   3   2   2   2   2   2   3
3   3   3   3   3   3   3   3
4   4   4   4   4   5   4   4   4   4   4   5
6   5   5   6   5   5   6   5   5   6   5   5
6   7
6   6   6   6   6
7   7   7   7   7   7   7   7   7
8   8   8   8   8   9   8   8   8   8   8   9
9   9   9   9   9   9   9   9
10  11  10  11  10  11  10  11  10  11 10  11  10  11  10  11  10  11  10  11
12  12  12  12  12  13  12  12  12  12  12  13
...

Crossrefs

See A187180-A187188 for alphabets of size 2 through 10.

See also A109337, A187199, A187200.

Sequence in context: A360964 A188794 A161966 * A358618 A183027 A359227

Adjacent sequences: A187185 A187186 A187187 * A187189 A187190 A187191

Keyword

nonn

Author

N. J. A. Sloane, Mar 06 2011

Status

approved