OFFSET
1,2
COMMENTS
a(n) is the number of bit strings of length n+5 with the pattern 01 at least thrice, and without the pattern 110, see example. - John M. Campbell, Jan 25 2013
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
FORMULA
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + x^2)/((1 - x - x^2)*(1 - x)^3).
a(n) = Fibonacci(n+8) - (21 + 10*n + 2*n^2). - G. C. Greubel, Jul 06 2019
EXAMPLE
From John M. Campbell, Jan 25 2013: (Start)
There are a(3) = 20 bit strings of length 3+5 with the pattern 01 at least thrice, and without the pattern 110:
00010101, 00100101, 00101001, 00101010, 00101011,
01000101, 01001001, 01001010, 01001011, 01010001,
01010010, 01010011, 01010100, 01010101, 01010111,
10010101, 10100101, 10101001, 10101010, 10101011.
(End)
MATHEMATICA
(See A213584.)
With[{F = Fibonacci}, Table[F[n+8] -(21+10*n+2*n^2), {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)
PROG
(PARI) vector(40, n, fibonacci(n+8) -(21+10*n+2*n^2)) \\ G. C. Greubel, Jul 06 2019
(Magma) [Fibonacci(n+8) -(21+10*n+2*n^2): n in [1..40]]; // G. C. Greubel, Jul 06 2019
(Sage) [fibonacci(n+8) -(21+10*n+2*n^2) for n in (1..40)] # G. C. Greubel, Jul 06 2019
(GAP) List([1..40], n-> Fibonacci(n+8) -(21+10*n+2*n^2)) # G. C. Greubel, Jul 06 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved