A057354 a(n) = floor(2*n/5).

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30

Offset: 0

Author

Mitch Harris

Keywords

• nonn
• easy

Comments

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

The sequence a(n) can be used in determining confidence intervals for the median of a population. Let Y(i) denote the i-th smallest datum in a random sample of size n from any population of values. When estimating the population median with a symmetric interval [Y(r), Y(n-r+1)], the exact confidence coefficient c for the interval is given by c = Sum_{k=r..n-r} C(n, k)(1/2)^n. If r = a(n-4), then the confidence coefficient will be (i) at least 0.90 for all n>=7, (ii) at least 0.95 for all n>=35, and (iii) at least 0.99 for all n>=115. To use the sequence, for example, decide on the minimum level of confidence desired, say 95%. Hence use a sample size of 35 or greater, say n=40. We then find a(n-4)=a(36)=14, and thus the 14th smallest and 14th largest values in the sample will form the bounds for the confidence interval. If the exact confidence coefficient c is needed, calculate c = Sum_{k=14..26} C(40,k)(1/2)^40, which is 0.9615226917. - Dennis P. Walsh, Nov 28 2011

a(n+2) is also the domination number of the n-antiprism graph. - Eric W. Weisstein, Apr 09 2016

Equals partial sums of 0 together with 0, 0, 1, 0, 1, ... (repeated). - Bruno Berselli, Dec 06 2016

Euler transform of length 5 sequence [1, 1, 0, -1, 1]. - Michael Somos, Dec 06 2016

Examples

			G.f. = x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + 4*x^11 + ...
		

References

• N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
• Dennis Walsh, Median estimation with the point-four-n-minus-two rule.
• Eric Weisstein's World of Mathematics, Antiprism Graph.
• Eric Weisstein's World of Mathematics, Domination Number.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A002266.

Programs

Magma

[2*n div 5: n in [0..80]]; // Bruno Berselli, Dec 06 2016

Mathematica

Table[Floor[2 n/5], {n, 0, 80}] (* Bruno Berselli, Dec 06 2016 *)
a[ n_] := Quotient[2 n, 5]; (* Michael Somos, Dec 06 2016 *)

PARI

a(n)=2*n\5 \\ Charles R Greathouse IV, Nov 28 2011

PARI

concat(vector(3), Vec(x^3*(1 + x^2) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^80))) \\ Colin Barker, Dec 06 2016.

Python

[int(2*n/5) for n in range(80)] # Bruno Berselli, Dec 06 2016

Sage

[floor(2*n/5) for n in range(80)] # Bruno Berselli, Dec 06 2016

Formula

G.f.: x^3*(1 + x^2) / ((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - Numerator corrected by R. J. Mathar, Feb 20 2011

a(n) = a(n-1) + a(n-5) - a(n-6) for n>5. - Colin Barker, Dec 06 2016

a(n) = -a(2-n) for all n in Z. - Michael Somos, Dec 06 2016

a(n) = A002266(2*n). - R. J. Mathar, Jul 21 2020

Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Sep 30 2022