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A002265
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Integers repeated 4 times.
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87
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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
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OFFSET
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0,9
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COMMENTS
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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
After initial terms, same as floor(n/2) - floor(n/4). - Jonathan Vos Post, Jan 19 2007
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
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REFERENCES
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V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.
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LINKS
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Todd Silvestri, Table of n, a(n) for n = 0..999
J. H. McClellan, 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).
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FORMULA
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a(n) = floor(n/4), n>=0;
a(n) = ( Sum_{k=0..n} (k+1)*cos(Pi*(n-k)/2}+1/4*(cos(n*Pi/2)+1+(-1)^n) )/2 - 1. - Paolo P. Lava, Oct 09 2006
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 + Sum_{k=0..n} ( (1/24)*( -5*(k mod 4) + ((k+1) mod 4) + ((k+2) mod 4) + 7*((k+3) mod 4)) ). - Paolo P. Lava, Jun 20 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
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
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MAPLE
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seq(seq(seq(k, i=2..3), j=2..3), k=0..19); # Zerinvary Lajos, Apr 29 2007
P:=proc(n) local a, i, k; for i from 0 by 1 to n do a:=-1+sum('1/24*(-5*(k mod 4)+((k+1) mod 4)+((k+2) mod 4)+7*((k+3) mod 4))', 'k'=0..i); print(a); od; end: P(100); # Paolo P. Lava, Jun 20 2007
A002265:=n->floor(n/4); seq(A002265(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013
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MATHEMATICA
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Table[Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
a[n_Integer/; n>=0]:=Quotient[2 n+(-1)^n+2 Mod[n^2 (3 n+2), 4, -1]+2 Mod[(n+1)^3, 4, -1]-3, 8] (* Todd Silvestri, Oct 27 2014 *)
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PROG
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(Sage) [floor(n/4) for n in xrange(0, 84)] # Zerinvary Lajos, Dec 02 2009
(PARI) a(n)=n\4 \\ Charles R Greathouse IV, Dec 10 2013
(MAGMA) [Floor(n/4): n in [0..80]]; // Vincenzo Librandi, Oct 28 2014
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CROSSREFS
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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.
a(n)+1 gives row 1 of the array in A238340.?
Sequence in context: A235492 A226762 A300763 * A242601 A110655 A008621
Adjacent sequences: A002262 A002263 A002264 * A002266 A002267 A002268
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Clarified my formulas Mohammad K. Azarian, Aug 01 2009
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STATUS
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approved
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