

A187180


Parse the infinite string 0101010101... into distinct phrases 0, 1, 01, 010, 10, ...; a(n) = length of nth phrase.


14



1, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38, 39, 38, 39, 40, 41, 40, 41, 42, 43, 42, 43, 44, 45, 44, 45, 46, 47, 46, 47, 48, 49, 48, 49, 50, 51, 50, 51, 52, 53, 52, 53, 54, 55, 54, 55, 56, 57, 56, 57, 58, 59, 58, 59, 60, 61
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OFFSET

1,3


LINKS

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,1).


FORMULA

Consider more generally the string 012...k012...k012...k012...k01... with an alphabet of size B, where k = B1. The sequence begins with B 1's, and thereafter is quasiperiodic with period B^2, and increases by B in each period.
For the present example, where B=2, the sequence begins with two 1's and thereafter increases by 2 in each block of 4: (1,1) (2,3,2,3), (4,5,4,5), (6,7,6,7), ...
From Colin Barker, Oct 15 2015: (Start)
a(n) = (1 + (1)^n + (1i)*(i)^n + (1+i)*i^n + 2*n) / 4 for n>1, where i = sqrt(1).
G.f.: x*(x^52*x^4+x^3+x^2+1) / ((x1)^2*(x+1)*(x^2+1)).
(End)


EXAMPLE

The sequence begins
1 1
2 3 2 3
4 5 4 5
6 7 6 7
8 9 8 9
10 11 10 11 ...


MAPLE

1, 1, seq(op(2*i*[1, 1, 1, 1]+[0, 1, 0, 1]), i=1..100); # Robert Israel, Oct 15 2015


MATHEMATICA

Join[{1}, LinearRecurrence[{1, 0, 0, 1, 1}, {1, 2, 3, 2, 3}, 119]] (* Ray Chandler, Aug 26 2015 *)
CoefficientList[Series[(x^5  2 x^4 + x^3 + x^2 + 1)/((x  1)^2 (x + 1) (x^2 + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Oct 16 2015 *)


PROG

(PARI) a(n) = if(n==1, 1, (1 + (1)^n + (1I)*(I)^n + (1+I)*I^n + 2*n) / 4); \\ Colin Barker, Oct 15 2015
(PARI) Vec(x*(x^52*x^4+x^3+x^2+1) / ((x1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 15 2015


CROSSREFS

See A187180A187188 for alphabets of size 2 through 10.
See also A109337, A187199, A187200, A106249, A083219, A018837.
Sequence in context: A083219 A106249 A110516 * A256992 A261323 A134986
Adjacent sequences: A187177 A187178 A187179 * A187181 A187182 A187183


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Mar 06 2011


STATUS

approved



