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A002266
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Integers repeated 5 times.
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45
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0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16
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OFFSET
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0,11
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COMMENTS
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For n > 3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000002 (see example). E.g., the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ...] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre, Jan 08 2006
Complement of A010874, since A010874(n) + 5*a(n) = n. - Hieronymus Fischer, Jun 01 2007
From Paul Curtz, May 13 2020: (Start)
Main N-S vertical of the pentagonal spiral built with this sequence is A001105:
21
20 15 15
20 14 10 10 15
20 14 9 6 6 10 15
20 14 9 5 3 3 6 10 15
20 14 9 5 2 1 1 3 6 10 16
19 14 9 5 2 0 0 0 1 3 6 11 16
19 13 9 5 2 0 0 1 3 7 11 16
19 13 8 5 2 2 1 4 7 11 16
19 13 8 4 4 4 4 7 11 16
19 13 8 8 8 7 7 11 17
18 13 12 12 12 12 12 17
18 18 18 18 17 17 17
The main S-N vertical and the next one are A000217. (End)
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LINKS
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Table of n, a(n) for n=0..82.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
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FORMULA
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a(n) = floor(n/5), n >= 0.
G.f.: x^5/((1-x)(1-x^5)).
a(n)= -1 + Sum_{k=0..n} ((8*(sin(2*Pi*k/5))^2-5)^2-5)/20, with n>=0. a(n)= -1 + Sum_{k=0..n} (1/50)*(-9*(k mod 5) + ((n+1) mod 5) + ((n+2) mod 5) + ((n+3) mod 5) + 11*((n+4) mod 5)), with n >= 0. - Paolo P. Lava, May 15 2007
a(n) = (n - A010874(n))/5. - Hieronymus Fischer, May 29 2007
Also, floor((n^5-1)/5*n^4) will produce this sequence. Moreover, floor((n^5-n^4)/(5*n^4-4*n^3)) (n >= 1) will produce this sequence as well. - Mohammad K. Azarian, Nov 08 2007
This sequence is also the sequence floor(n*e^(-(1+sqrt(5))/2))(n >= 1). - Mohammad K. Azarian, May 13 2008
For n >= 5, a(n) = floor(log_5(5^a(n-1) + 5^a(n-2) + 5^a(n-3) + 5^a(n-4) + 5^a(n-5))). - Vladimir Shevelev, Jun 22 2010
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MAPLE
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A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013
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MATHEMATICA
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Table[Floor[n/5], {n, 0, 20}] (* Wesley Ivan Hurt, Dec 10 2013 *)
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PROG
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(Sage) [floor(n/5) - 1 for n in range(5, 88)] # Zerinvary Lajos, Dec 01 2009
(Haskell)
a002266 = (`div` 5)
a002266_list = [0, 0, 0, 0, 0] ++ map (+ 1) a002266_list
-- Reinhard Zumkeller, Nov 27 2012
(PARI) a(n)=n\5 \\ Charles R Greathouse IV, Dec 10 2013
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CROSSREFS
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a(n) = A010766(n, 5).
Cf. A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A004526, A010872, A010873, A010874.
Partial sums: A130520.
Sequence in context: A092278 A105512 A301506 * A075249 A008648 A154099
Adjacent sequences: A002263 A002264 A002265 * A002267 A002268 A002269
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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