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%I A002266 #71 Dec 14 2023 05:57:53
%S A002266 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,
%T A002266 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,
%U A002266 12,12,12,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16
%N A002266 Integers repeated 5 times.
%C A002266 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 A000032 (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
%C A002266 Complement of A010874, since A010874(n) + 5*a(n) = n. - _Hieronymus Fischer_, Jun 01 2007
%C A002266 From _Paul Curtz_, May 13 2020: (Start)
%C A002266 Main N-S vertical of the pentagonal spiral built with this sequence is A001105:
%C A002266                               21
%C A002266                           20  15  15
%C A002266                       20  14  10  10  15
%C A002266                   20  14   9   6   6  10  15
%C A002266               20  14   9   5   3   3   6  10  15
%C A002266           20  14   9   5   2   1   1   3   6  10  16
%C A002266       19  14   9   5   2   0   0   0   1   3   6  11  16
%C A002266         19   13   9   5  2   0   0    1   3   7 11  16
%C A002266           19  13    8  5   2   2   1   4   7  11  16
%C A002266             19   13   8  4   4   4   4   7  11  16
%C A002266               19  13   8   8   8   7   7  11  17
%C A002266                  18  13  12 12  12  12  12  17
%C A002266                    18  18 18  18  17  17  17
%C A002266 The main S-N vertical and the next one are A000217. (End)
%H A002266 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A002266 a(n) = floor(n/5), n >= 0.
%F A002266 G.f.: x^5/((1-x)(1-x^5)).
%F A002266 a(n) = (n - A010874(n))/5. - _Hieronymus Fischer_, May 29 2007
%F A002266 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
%F A002266 Sum_{n>=5} (-1)^(n+1)/a(n) = log(2) (A002162). - _Amiram Eldar_, Sep 30 2022
%p A002266 A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # _Wesley Ivan Hurt_, Dec 10 2013
%t A002266 Table[Floor[n/5], {n,0,20}] (* _Wesley Ivan Hurt_, Dec 10 2013 *)
%t A002266 Table[{n,n,n,n,n},{n,0,20}]//Flatten (* _Harvey P. Dale_, Jun 17 2022 *)
%o A002266 (Sage) [floor(n/5) - 1 for n in range(5,88)] # _Zerinvary Lajos_, Dec 01 2009
%o A002266 (Haskell)
%o A002266 a002266 = (`div` 5)
%o A002266 a002266_list = [0,0,0,0,0] ++ map (+ 1) a002266_list
%o A002266 -- _Reinhard Zumkeller_, Nov 27 2012
%o A002266 (PARI) a(n)=n\5 \\ _Charles R Greathouse IV_, Dec 10 2013
%o A002266 (Python)
%o A002266 def A002266(n): return n//5 # _Chai Wah Wu_, Nov 08 2022
%Y A002266 a(n) = A010766(n, 5).
%Y A002266 Cf. A002162, A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A010872, A010873, A010874.
%Y A002266 Partial sums: A130520.
%K A002266 nonn,easy
%O A002266 0,11
%A A002266 _N. J. A. Sloane_
%E A002266 Incorrect formula removed by _Ridouane Oudra_, Oct 16 2021

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