A173562 -id:A173562 - 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)

A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).

Original entry on oeis.org

0, 1, 5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, 1701, 1785, 1870, 1958, 2047, 2139, 2232, 2328, 2425, 2525, 2626

Offset: 0

N. J. A. Sloane

Maximum value of Voronoi's principal quadratic form of the first type when variables restricted to {-1,0,1}. - Michael Somos, Mar 10 2004
This is the main row of a version of the "square spiral" when read alternatively from left to right (see link). See also A001107, A007742, A033954, A033991. It is easy to see that the only prime in the sequence is 5. - Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 08 2009
From Mitch Phillipson, Manda Riehl, Tristan Williams, Mar 06 2009: (Start)
a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 12, using the following ordering:
In S_j, a permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b. We extend this notion to S_j \wr C_n as follows. Element psi =[ alpha_1^beta_1, ... alpha_j^beta_j ] avoids tau = [ a_1 ... a_m ] (tau in S_m) if psi' = [ alpha_1*beta_1 ... alpha_j*beta_j ] avoids tau in the usual sense. For n=2, there are 5 elements of S_2 \wr C_2 that avoid the pattern 12. They are: [ 2^1,1^1 ], [ 2^2,1^1 ], [ 2^2,1^2 ], [ 2^1,1^2 ], [ 1^2,2^1 ].
For example, if psi = [2^1,1^2], then psi'=[2,2] which avoids tau=[1,2] because no subsequence ab of psi' has a < b. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 115.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Emilio Apricena, A version of the Ulam spiral (This is called the square spiral. - _N. J. A. Sloane_, Jul 27 2018)
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), see p. 283.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Partial sums of A042948.
Cf. A002265, A004526, A006578, A011848, A133983, A139592, A173562, A253146.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[n^2 + n - 1 - Floor((n-1)/2): n in [0..25]]; // G. C. Greubel, Oct 29 2017

A035608:=n->floor((n + 1/4)^2): seq(A035608(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017

Table[n^2 + Floor[n/2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
CoefficientList[Series[x (1 + 3 x)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 5, 10}, 60] (* Harvey P. Dale, Feb 21 2013 *)

PARI
```
a(n)=n^2+n-1-(n-1)\2
```

a(n) = n^2 + n - 1 - floor((n-1)/2).
a(n) = A011848(2*n+1).
a(n) = A002378(n) - A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*A006578(n) - A002378(n)/2 = A139592(n)/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = A002265(n+2) + A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Jan 27 2010
a(n) = (-1)^n*Sum_{i=0..n} (-1)^i*(2*i^2 + 3*i + 1). Omits the leading 0. - William A. Tedeschi, Aug 25 2010
a(n) = n^2 + floor(n/2), from Mathematica section. - Vladimir Joseph Stephan Orlovsky, Apr 12 2011
a(0)=0, a(1)=1, a(2)=5, a(3)=10; for n > 3, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Feb 21 2013
For n > 1: a(n) = a(n-2) + 4*n - 3; see also row sums of triangle A253146. - Reinhard Zumkeller, Dec 27 2014
a(n) = 3*A002620(n) + A002620(n+1). - R. J. Mathar, Jul 18 2015
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 4 - 2*log(2) - Pi/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/3 - 4*(1-log(2)). (End)
E.g.f.: (x*(2*x + 3)*cosh(x) + (2*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2024

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

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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A035608 Expansion of g.f. x*(1 + 3*x)/((1 + x)*(1 - x)^3).

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

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

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A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).