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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.
2
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
1,7
COMMENTS
See A187180 for details.
LINKS
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).
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)
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 *)
PROG
(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
CROSSREFS
See A187180-A187188 for alphabets of size 2 through 10.
Sequence in context: A125604 A216685 A331853 * A301375 A325273 A352541
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 06 2011
STATUS
approved