A002266 Integers repeated 5 times.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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)

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

a(n) = A010766(n, 5).

Cf. A002162, A008648, A004526, A002264, A002265, A010761, A010762, A110532, A110533, A010872, A010873, A010874.

Partial sums: A130520.

Programs

Haskell

a002266 = (`div` 5)
a002266_list = [0,0,0,0,0] ++ map (+ 1) a002266_list
-- Reinhard Zumkeller, Nov 27 2012

Maple

A002266:=n->floor(n/5); seq(A002266(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Mathematica

Table[Floor[n/5], {n,0,20}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Jun 17 2022 *)

PARI

a(n)=n\5 \\ Charles R Greathouse IV, Dec 10 2013

Python

def A002266(n): return n//5 # Chai Wah Wu, Nov 08 2022

Sage

[floor(n/5) - 1 for n in range(5,88)] # Zerinvary Lajos, Dec 01 2009

Formula

a(n) = floor(n/5), n >= 0.

G.f.: x^5/((1-x)(1-x^5)).

a(n) = (n - A010874(n))/5. - Hieronymus Fischer, May 29 2007

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

Sum_{n>=5} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Extensions

Incorrect formula removed by Ridouane Oudra, Oct 16 2021