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A187185
Parse the infinite string 0123456012345601234560... into distinct phrases 0, 1, 2, 3, 4, 5, 6, 01, 23, 45, 60, 12, 34, 56, 012, ...; a(n) = length of n-th phrase.
2
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 15, 14, 15, 14, 15, 14, 15, 14, 15, 14, 15, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18
OFFSET
1,8
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
FORMULA
After the initial block of seven 1's, the sequence is quasi-periodic with period 49, increasing by 7 after each block.
From Colin Barker, Jan 31 2020: (Start)
G.f.: x*(1 + x^7 + x^14 + x^21 + x^28 + x^35 + x^42 + x^43 - x^44 + x^45 - x^46 + x^47 - x^48 - x^50 + x^51 - x^52 + x^53 - x^54 + x^55) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 + x^7 + x^14 + x^21 + x^28 + x^35 + x^42)).
a(n) = a(n-1) + a(n-49) - a(n-50) for n>56.
(End)
MATHEMATICA
Join[{1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8}, 114]] (* Ray Chandler, Aug 26 2015 *)
PROG
(PARI) Vec(x*(1 + x^7 + x^14 + x^21 + x^28 + x^35 + x^42 + x^43 - x^44 + x^45 - x^46 + x^47 - x^48 - x^50 + x^51 - x^52 + x^53 - x^54 + x^55) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)*(1 + x^7 + x^14 + x^21 + x^28 + x^35 + x^42)) + O(x^80)) \\ Colin Barker, Jan 31 2020
CROSSREFS
See A187180-A187188 for alphabets of size 2 through 10.
Sequence in context: A133877 A132270 A195174 * A054896 A052364 A052374
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 06 2011
STATUS
approved