A187183 Parse the infinite string 012340123401234012340... into distinct phrases 0, 1, 2, 3, 4, 01, 23, 40, 12, ...; a(n) = length of n-th phrase.

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

Offset: 1

Author

N. J. A. Sloane, Mar 06 2011

Keywords

• nonn
• easy

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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Crossrefs

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

Programs

Mathematica

Join[{1, 1, 1, 1},LinearRecurrence[{1, 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},{1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6},96]] (* Ray Chandler, Aug 26 2015 *)

PARI

Vec(x*(1 + x^5 + x^10 + x^15 + x^20 + x^21 - x^22 + x^23 - x^24 - x^26 + x^27 - x^28 + x^29) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)*(1 + x^5 + x^10 + x^15 + x^20)) + O(x^80)) \\ Colin Barker, Jan 31 2020

Formula

After the initial block of five 1's, the sequence is quasi-periodic with period 25, increasing by 5 after each block.

From Colin Barker, Jan 31 2020: (Start)

G.f.: x*(1 + x^5 + x^10 + x^15 + x^20 + x^21 - x^22 + x^23 - x^24 - x^26 + x^27 - x^28 + x^29) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)*(1 + x^5 + x^10 + x^15 + x^20)).

a(n) = a(n-1) + a(n-25) - a(n-26) for n>30.

(End)