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
Examples
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 ...
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).
Programs
-
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 *) -
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
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)
Comments