A057353 -id:A057353 - cp's OEIS frontend

A002265 Nonnegative integers repeated 4 times.

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

N. J. A. Sloane

For n>=1 and i=sqrt(-1) let F(n) the n X n matrix of the Discrete Fourier Transform (DFT) whose element (j,k) equals exp(-2*Pi*i*(j-1)*(k-1)/n)/sqrt(n). The multiplicities of the four eigenvalues 1, i, -1, -i of F(n) are a(n+4), a(n-1), a(n+2), a(n+1), hence a(n+4) + a(n-1) + a(n+2) + a(n+1) = n for n>=1. E.g., the multiplicities of the eigenvalues 1, i, -1, -i of the DFT-matrix F(4) are a(8)=2, a(3)=0, a(6)=1, a(5)=1, summing up to 4. - Franz Vrabec, Jan 21 2005
Complement of A010873, since A010873(n)+4*a(n)=n. - Hieronymus Fischer, Jun 01 2007
For even values of n, a(n) gives the number of partitions of n into exactly two parts with both parts even. - Wesley Ivan Hurt, Feb 06 2013
a(n-4) counts number of partitions of (n) into parts 1 and 4. For example a(11) = 3 with partitions (44111), (41111111), (11111111111). - David Neil McGrath, Dec 04 2014
a(n-4) counts walks (closed) on the graph G(1-vertex; 1-loop, 4-loop) where order of loops is unimportant. - David Neil McGrath, Dec 04 2014
Number of partitions of n into 4 parts whose smallest 3 parts are equal. - Wesley Ivan Hurt, Jan 17 2021

V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.

Todd Silvestri, Table of n, a(n) for n = 0..999
J. H. McClellan and T. W. Parks, Eigenvalue and Eigenvector Decomposition of the Discrete Fourier Transform, IEEE Trans. Audio and Electroacoust., Vol. AU-20, No. 1, March 1972, pp. 66-74.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008615, A008621, A249356.
Zero followed by partial sums of A011765.
Partial sums: A130519. Other related sequences: A004526, A010872, A010873, A010874.
Third row of A180969.

[Floor(n/4): n in [0..80]]; // Vincenzo Librandi, Oct 28 2014

A002265:=n->floor(n/4); seq(A002265(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Aug 08 2020 *)

a(n)=n\4 \\ Charles R Greathouse IV, Dec 10 2013

def A002265(n): return n>>2 # Chai Wah Wu, Jul 27 2022

[floor(n/4) for n in range(0,84)] # Zerinvary Lajos, Dec 02 2009

a(n) = floor(n/4), n>=0;
G.f.: (x^4)/((1-x)*(1-x^4)).
a(n) = (2*n-(3-(-1)^n-2*(-1)^floor(n/2)))/8; also a(n) = (2*n-(3-(-1)^n-2*sin(Pi/4*(2*n+1+(-1)^n))))/8 = (n-A010873(n))/4. - Hieronymus Fischer, May 29 2007
a(n) = (1/4)*(n-(3-(-1)^n-2*(-1)^((2*n-1+(-1)^n)/4))/2). - Hieronymus Fischer, Jul 04 2007
a(n) = floor((n^4-1)/4*n^3) (n>=1); a(n) = floor((n^4-n^3)/(4*n^3-3*n^2)) (n>=1). - Mohammad K. Azarian, Nov 08 2007 and Aug 01 2009
For n>=4, a(n) = floor( log_4( 4^a(n-1) + 4^a(n-2) + 4^a(n-3) + 4^a(n-4) ) ). - Vladimir Shevelev, Jun 22 2010
a(n) = A180969(2,n). - Adriano Caroli, Nov 26 2010
a(n) = A173562(n)-A000290(n); a(n+2) = A035608(n)-A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n+1) = A140201(n) - A057353(n+1). - Reinhard Zumkeller, Feb 26 2011
a(n) = ceiling((n-3)/4), n >= 0. - Wesley Ivan Hurt, Jun 01 2013
a(n) = (2*n + (-1)^n + 2*sin(Pi*n/2) + 2*cos(Pi*n/2) - 3)/8. - Todd Silvestri, Oct 27 2014
E.g.f.: (x/4 - 3/8)*exp(x) + exp(-x)/8 + (sin(x)+cos(x))/4. - Robert Israel, Oct 30 2014
a(n) = a(n-1) + a(n-4) - a(n-5) with initial values a(3)=0, a(4)=1, a(5)=1, a(6)=1, a(7)=1. - David Neil McGrath, Dec 04 2014
a(n) = A004526(A004526(n)). - Bruno Berselli, Jul 01 2016
From Guenther Schrack, May 03 2019: (Start)
a(n) = (2*n - 3 + (-1)^n + 2*(-1)^(n*(n-1)/2))/8.
a(n) = a(n-4) + 1, a(k)=0 for k=0,1,2,3, for n > 3. (End)

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Jaume Oliver Lafont, Oct 15 2009

			G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Characteristic function of A042968, whose complement A008586 gives the positions of zeros (after its initial term).
Absolute values of A046978, A075553, A131729, A358839, and for n >= 1, also of A112299 and of A257196.
Sequence A152822 shifted by two terms.
Parity of A327860, A331733, A342002, A351083 and A345000.
Row 3 of A225145, Column 2 of A229940 (after the initial term).
First differences of A057353. Sum of A359370 and A359372.
Cf. A000035, A011655, A011558, A097325, A109720, A168181, A168182, A168184, A145568, A168185 (characteristic functions for numbers that are not multiples of k = 2, 3 and 5..12).
Cf. A016628, A164985, A165132.
Cf. A010873, A033436, A069733 (inverse Möbius transform), A121262 (one's complement), A190621 [= n*a(n)], A355689 (Dirichlet inverse).

[Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

PadRight[{},120,{0,1,1,1}] (* Harvey P. Dale, Jul 04 2013 *)
Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)
a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PARI
```
{a(n) = !!(n%4)};
```

def A166486(n): return (0,1,1,1)[n&3] # Chai Wah Wu, Jan 03 2023

G.f.: (x + x^2 + x^3) / (1 - x^4) = x * (1 + x + x^2) / ((1 - x) * (1 + x) * (1 + x^2)) = x * (1 - x^3) / ((1 - x) * (1 - x^4)).
a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).
Sum_{k>0} a(k)/(k*3^k) = log(5)/4.
From Reinhard Zumkeller, Nov 30 2009: (Start)
Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0.
a(n) = 1-A121262(n).
a(A042968(n))=1; a(A008586(n))=0.
A033436(n) = Sum{k=0..n} a(k)*(n-k). (End)
a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010
a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010
Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011
Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011
a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011
a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13 2012
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
For n >= 1, a(n) = A053866(A225546(n)) = A000035(A331733(n)). - Antti Karttunen, Jul 07 2020
a(n) = signum(n mod 4). - Alois P. Heinz, May 12 2021
From Antti Karttunen, Dec 28 2022: (Start)
a(n) = [A010873(n) > 0], where [ ] is the Iverson bracket.
a(n) = abs(A046978(n)) = abs(A075553(n)) = abs(A131729(n)) = abs(A358839(n)).
For all n >= 1, a(n) = abs(A112299(n)) = abs(A257196(n))
a(n) = A152822(2+n).
a(n) = A359370(n) + A359372(n). (End)
E.g.f.: (cosh(x) - cos(x))/2 + sinh(x). - Stefano Spezia, Aug 04 2025

Secondary definition (from Reinhard Zumkeller's Nov 30 2009 comment) added to the name by Antti Karttunen, Dec 20 2022

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

Original entry on oeis.org

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

Mitch Harris

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

			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 + ...

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.

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).

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

Cf. A002266.

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

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

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

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.

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

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

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

A057357 a(n) = floor(3*n/7).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 0

Mitch Harris

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

This sequence relates to 3/7 = 0.42857142... (essentially given by A020806). It differs from the Beatty sequence A308358 for sqrt(3)/4 = 0.43301270... = A120011.

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.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
Index to sequences related to Beatty sequences.

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A020806, A193535, A308358.

[Floor(3*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[(3*Range[0,80])/7] (* Harvey P. Dale, Aug 22 2013 *)

a(n)=3*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x^2+x^4)*x^3/((1-x)*(1-x^7)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
for all m>=0 a(7m)=0 mod 3; a(7m+1)=0 mod 3; a(7m+2)= 0 mod 3; a(7m+3) = 1 mod 3; a(5m+4) = 1 mod 3; a(7m+5) = 2 mod 3; a(7m+6) = 2 mod 3 - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/3 (A193535). - Amiram Eldar, Sep 30 2022

A057355 a(n) = floor(3*n/5).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42, 43, 43

Offset: 0

Mitch Harris

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

The sequence can be obtained from A008588 by deleting the last digit of each term. - Bruno Berselli, Sep 11 2019

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.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A008588.

[3*n div 5: n in [0..80]]; // Bruno Berselli, Dec 07 2016

Table[Floor[3*n/5], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n)=3*n\5 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: x^2*(1 + x^2 + x^3)/((1 - x)*(1 - x^5)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
For all m>=0: a(5m)=0 mod 3; a(5m+1)=0 mod 3; a(5m+2)=1 mod 3; a(5m+3)=1 mod 3; a(5m+4)=2 mod 3.
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Sep 30 2022

A057356 a(n) = floor(2*n/7).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22

Offset: 0

Mitch Harris

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

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A003881.

[Floor(2*n/7): n in [0..50]]; // G. C. Greubel, Nov 03 2017

Table[Floor[2*n/7], {n,0,50}] (* G. C. Greubel, Nov 03 2017 *)

a(n)=2*n\7 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x^4*(1+x)*(x^2-x+1)/( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - Numerator corrected by R. J. Mathar, Feb 20 2011

Sum_{n>=4} (-1)^n/a(n) = Pi/4 (A003881). - Amiram Eldar, Sep 30 2022

A057359 a(n) = floor(5*n/7).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47, 48, 49, 50, 50, 51

Offset: 0

Mitch Harris

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

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.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A001622.

[Floor(5*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[5 Range[0,75]/7]  (* Harvey P. Dale, Mar 18 2011 *)

a(n)=5*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f. x^2*(1+x+x^3+x^4+x^5) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). Numerator corrected Feb 20 2011

Sum_{n>=2} (-1)^n/a(n) = sqrt(10-2*sqrt(5))*Pi/10 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 30 2022

A057367 a(n) = floor(11*n/30).

Original entry on oeis.org

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

Offset: 0

Mitch Harris

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

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.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
Index entries for sequences related to Beatty sequences

Similar pattern in Islamic leap years A057347. Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(11*n/30): n in [0..50]]; // G. C. Greubel, Nov 03 2017

A057367:=n->floor(11*n/30); seq(A057367(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[11n/30], {n,0,100}] (* Wesley Ivan Hurt, Oct 29 2013 *)

a(n)=11*n\30 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-30) - a(n-31).

G.f.: x^3*(1 + x^3 + x^6 + x^8 + x^11 + x^14 + x^17 + x^19 + x^22 + x^25 + x^27)/( (1+x)*(1+x+x^2)*(x^2-x+1)*(x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)*(x^8+x^7-x^5-x^4-x^3+x+1)*(x-1)^2 ). [Corrected by R. J. Mathar, Feb 20 2011]

A057358 a(n) = floor(4*n/7).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 38, 39, 40, 40, 41, 41, 42

Offset: 0

Mitch Harris

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

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.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1)

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(4*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Table[Floor[4*n/7], {n, 0, 50}] (* G. C. Greubel, Nov 02 2017 *)

a(n)=4*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f. x^2*(1+x^2+x^4+x^5) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ) - Numerator corrected by R. J. Mathar, Feb 20 2011

Sum_{n>=2} (-1)^n/a(n) = (Pi - 2*log(sqrt(2)+1))/(4*sqrt(2)). - Amiram Eldar, Sep 30 2022

A057360 a(n) = floor(3*n/8).

Original entry on oeis.org

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

Offset: 0

Mitch Harris

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

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A008585, A032766, A132292.

[Floor(3*n/8): n in [0..80]]; // Vincenzo Librandi, Jul 07 2011

A057360:=n->floor(3*n/8): seq(A057360(n), n=0..100); # Wesley Ivan Hurt, May 15 2015

Floor[3 Range[0, 100]/8] (* Wesley Ivan Hurt, May 15 2015 *)
LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{0,0,0,1,1,1,2,2,3},80] (* Harvey P. Dale, Jan 10 2025 *)

a(n)=3*n>>3 \\ Charles R Greathouse IV, Jul 07 2011

G.f.: x^3*(1+x^3+x^5) / ( (1+x)*(x^2+1)*(x^4+1)*(x-1)^2 ).
From Wesley Ivan Hurt, May 15 2015: (Start)
a(n) = a(n-1)+a(n-8)-a(n-9).
a(n) = A132292(A008585(n)), n>0.
a(n) = A002265(A032766(n)). (End)
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Sep 30 2022

Numerator of g.f. corrected by R. J. Mathar, Feb 20 2011

cp's OEIS Frontend

A002265 Nonnegative integers repeated 4 times.

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A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

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

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A057357 a(n) = floor(3*n/7).

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A057355 a(n) = floor(3*n/5).

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A057356 a(n) = floor(2*n/7).

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