A140201 -id:A140201 - 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)

A057353 a(n) = floor(3n/4).

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 54

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.
For n >= 2, a(n) is the number of different integers that can be written as floor(k^2/n) for k = 1, 2, 3, ..., n-1. Generalization of the 1st problem proposed during the 15th Balkan Mathematical Olympiad in 1998 where the question was asked for n = 1998 with a(1998) = 1498. - Bernard Schott, Apr 22 2022
For n > 1, a(n) is also the Hadwiger number of the (n+1)-cycle complement graph (up to at least n = 16). - Eric W. Weisstein, Mar 10 2025

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
Balkan Mathematical Olympiad, Problem 1, 15th Balkan Mathematical Olympiad 1998.
Eric Weisstein's World of Mathematics, Cycle Complement Graph.
Eric Weisstein's World of Mathematics, Hadwiger Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences related to Beatty sequences.
Index to sequences related to Olympiads.

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

Cf. A002378, A007310, A057077, A073010, A173562, A182210.

[Floor(3*n/4): n in [0..90]]; // Vincenzo Librandi, Feb 12 2012

Table[Floor[3 n/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Floor[3 Range[0, 20]/4] (* Eric W. Weisstein, Mar 10 2025 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 3, 3}, {0, 20}] (* Eric W. Weisstein, Mar 10 2025 *)
CoefficientList[Series[x^2 (1 + x + x^2)/(1 - x - x^4 + x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 10 2025 *)

a(n)=3*n\4 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3
a(n) = A002378(n) - A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n+1) = A140201(n) - A002265(n+1). - Reinhard Zumkeller, Jan 26 2011
a(n) = n-1 - A002265(n-1) = ( A007310(n) + A057077(n+1) )/4 for n>0. a(n) = a(n-1)+a(n-4)-a(n-5) for n>4. - Bruno Berselli, Jan 28 2011
a(n) = 1/8*(6*n + 2*cos((Pi*n)/2) + cos(Pi*n) - 2*sin((Pi*n)/2) - 3). - Ilya Gutkovskiy, Sep 18 2015
a(4n) = a(4n+1). - Altug Alkan, Sep 26 2015
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Sep 29 2022

A140081 Period 4: repeat [0, 1, 1, 2].

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1

Offset: 0

Nadia Heninger and N. J. A. Sloane, Jun 03 2008

Also fix e = 4; then a(n) = minimal Hamming distance between the binary representation of n and the binary representation of any multiple k*e (0 <= k <= n/e) which is a child of n.

A number m is a child of n if the binary representation of n has a 1 in every position where the binary representation of m has a 1.

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

Cf. A140201. - Reinhard Zumkeller, Feb 21 2010

a140081 n = div (mod n 4 + mod n 2) 2
a140081_list = cycle [0, 1, 1, 2]  -- Reinhard Zumkeller, Aug 15 2015

A140081:=n->floor((3*(n mod 4)+1)/4); seq(A140081(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014

PadLeft[{}, 100, {0,1,1,2}] (* Harvey P. Dale, Aug 19 2011 *)
Table[Floor[(3 Mod[n, 4] + 1)/4], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 27 2014 *)

a(n)=n%4-n%4\2 \\ Jaume Oliver Lafont, Aug 28 2009

a(n) = 1 + a(n - 1 - a(n-1)) + 2*a(a(n-1)) - 2*a(n-1), a(0)=0. - Ramasamy Chandramouli, Jan 31 2010
a(n) = A047624(n+2) - A047624(n+1) - 1. - Reinhard Zumkeller, Feb 21 2010
a(n) = 1 - cos(Pi*n/2)/2 - sin(Pi*n/2)/2 - (-1)^n/2. - R. J. Mathar, Oct 08 2011
a(n) = ((n mod 4) + (n mod 2))/2. - Gary Detlefs, Apr 21 2012
From Colin Barker, Jan 13 2013: (Start)
a(n) = a(n-4).
G.f.: -x*(2*x^2+x+1) / ((x-1)*(x+1)*(x^2+1)). (End)
a(n) = floor((3*(n mod 4) + 1)/4). - Wesley Ivan Hurt, Mar 27 2014
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) = floor(1/2 + (n mod 4)/2).
a(n) = 1 - (-1)^n/2 - (-1)^(n/2 - 1/4 + (-1)^n/4)/2. (End)
a(n) = n - floor(n/2) - 2*floor(n/4). - Ridouane Oudra, Oct 30 2019

A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 11, 13, 16, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 45, 48, 49, 51, 53, 56, 57, 59, 61, 64, 65, 67, 69, 72, 73, 75, 77, 80, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 112, 113, 115, 117, 120, 121, 123

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A004526, A016813, A042948, A047470, A140081, A140201, A173562.

[n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // Wesley Ivan Hurt, Jun 01 2016

A047624:=n->(8*n-11-I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047624(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* G. C. Greubel, Jun 01 2016 *)

From Reinhard Zumkeller, Feb 21 2010: (Start)
a(n+1) = A173562(n) - A173562(n-1);
a(n+1) - a(n) = A140081(n-1) + 1;
a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)
G.f.: x^2*(1+2*x+2*x^2+3*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)
E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 20 2021

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A002265 Nonnegative integers repeated 4 times.

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A057353 a(n) = floor(3n/4).

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A140081 Period 4: repeat [0, 1, 1, 2].

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A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

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