A187184 Parse the infinite string 0123450123450123450... into distinct phrases 0, 1, 2, 3, 4, 5, 01, 23, 45, 012, 34, ...; a(n) = length of n-th phrase.
1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 4, 3, 3, 4, 3, 3, 4, 5, 4, 4, 4, 5, 5, 5, 5, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 8, 8, 9, 8, 8, 8, 9, 10, 9, 9, 10, 9, 9, 10, 11, 10, 10, 10, 11, 11, 11, 11, 11, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 14, 14, 14, 15, 14, 14, 14, 15, 16, 15, 15, 16, 15, 15, 16, 17, 16, 16, 16, 17, 17, 17, 17, 17, 18, 19, 18, 19, 18, 19, 18, 19, 18, 19, 18, 19, 20, 20, 20, 21, 20, 20
Offset: 1
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
Programs
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Mathematica
Join[{1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1},{1, 2, 2, 2, 3, 2, 2, 2, 3, 4, 3, 3, 4, 3, 3, 4, 5, 4, 4, 4, 5, 5, 5, 5, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7},115]] (* Ray Chandler, Aug 26 2015 *) -
PARI
Vec(x*(1 + x^6 + x^9 - x^10 + x^13 + x^14 - x^15 + x^17 - x^18 + x^20 + x^21 - x^22 + x^25 + x^30 + x^31 - x^32 + x^33 - x^34 + x^35 - 2*x^36 + x^37 - x^38 + x^39 - x^40 + x^41) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)*(1 - x^3 + x^6)*(1 + x^3 + x^6)*(1 - x^6 + x^12)) + O(x^80)) \\ Colin Barker, Jan 31 2020
Formula
After the initial block of six 1's, the sequence is quasi-periodic with period 36, increasing by 6 after each block.
From Colin Barker, Jan 31 2020: (Start)
G.f.: x*(1 + x^6 + x^9 - x^10 + x^13 + x^14 - x^15 + x^17 - x^18 + x^20 + x^21 - x^22 + x^25 + x^30 + x^31 - x^32 + x^33 - x^34 + x^35 - 2*x^36 + x^37 - x^38 + x^39 - x^40 + x^41) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)*(1 - x^3 + x^6)*(1 + x^3 + x^6)*(1 - x^6 + x^12)).
a(n) = a(n-1) + a(n-36) - a(n-37) for n>42.
(End)
Comments