A187182 Parse the infinite string 0123012301230123... into distinct phrases 0, 1, 2, 3, 01, 23, 012, ...; a(n) = length of n-th phrase.

1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 5, 4, 5, 4, 5, 4, 5, 6, 6, 7, 6, 6, 7, 7, 7, 8, 9, 8, 9, 8, 9, 8, 9, 10, 10, 11, 10, 10, 11, 11, 11, 12, 13, 12, 13, 12, 13, 12, 13, 14, 14, 15, 14, 14, 15, 15, 15, 16, 17, 16, 17, 16, 17, 16, 17, 18, 18, 19, 18, 18, 19, 19, 19, 20, 21, 20, 21, 20, 21, 20, 21, 22, 22, 23, 22, 22, 23, 23, 23, 24, 25, 24, 25, 24, 25, 24, 25

Offset

1,5

Comments

See A187180 for details.

Links

Ray Chandler, 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, 1, -1).

Formula

From Colin Barker, Jan 31 2020: (Start)

G.f.: x*(1 + x^4 + x^6 - x^7 + x^9 + x^12 + x^13 - x^14 + x^15 - 2*x^16 + x^17 - x^18 + x^19) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)).

a(n) = a(n-1) + a(n-16) - a(n-17) for n>20.

(End)

Example

The sequence is quasi-periodic with period 16, increasing by 4 after each block:
1   1   1   1
2   2   3   2   2   3   3   3   4   5   4   5   4   5   4   5
6   6   7   6   6   7   7   7   8   9   8   9   8   9   8   9
10  10  11  10  10  11  11  11  12  13  12  13  12  13  12  13
14  14  15  14  14  15  15  15  16  17  16  17  16  17  16  17
...

Mathematica

Join[{1, 1, 1}, LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 5, 4, 5, 4, 5, 4, 5}, 97]] (* Ray Chandler, Aug 26 2015 *)

Prog

(PARI) Vec(x*(1 + x^4 + x^6 - x^7 + x^9 + x^12 + x^13 - x^14 + x^15 - 2*x^16 + x^17 - x^18 + x^19) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)) + O(x^80)) \\ Colin Barker, Jan 31 2020

Crossrefs

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

Sequence in context: A334796 A140361 A237769 * A362816 A176208 A375422

Adjacent sequences: A187179 A187180 A187181 * A187183 A187184 A187185

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 06 2011

Status

approved