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A002265
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Integers repeated 4 times.
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65
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| 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 (franz.vrabec(AT)aon.at), Jan 21 2005
After initial terms, same as [n/2] - [n/4]. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 19 2007
Complement of A010873, since A010873(n)+4*a(n)=n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
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REFERENCES
| V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.
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.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| 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 (paoloplava(AT)gmail.com), Oct 09 2006
G.f.: (x^4)/((1-x)*(1-x^4))
a(n)=(2n-(3-(-1)^n-2*(-1)^floor(n/2)))/8; also a(n)=(2n-(3-(-1)^n-2*sin(pi/4*(2n+1+(-1)^n))))/8=(n-A010873(n))/4. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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 (paoloplava(AT)gmail.com), Jun 20 2007
a(n)=(1/4)*(n-(3-(-1)^n-2*(-1)^((2n-1+(-1)^n)/4))/2). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jul 04 2007
Also, floor((n^4-1)/4n^3) (n>=1) will produce this sequence. Moreover, floor((n^4-n^3)/(4n^3-3n^2)) (n>=1) will produce this sequence as well. - Mohammad K. Azarian (azarian(AT)evansville.edu), 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))) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 22 2010]
a(n)=A180969(2,n) - Adriano Caroli (adriano_caroli(AT)virgilio.it), Nov 26 2010
a(n) = A173562(n)-A000290(n); a(n+2) = A035608(n)-A173562(n). [From Reinhard Zumkeller, Feb 21 2010]
a(n+1) = A140201(n) - A057353(n+1). [From Reinhard Zumkeller, Feb 26 2011]
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MAPLE
| seq(seq(seq(k, i=2..3), j=2..3), k=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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 (paoloplava(AT)gmail.com), Jun 20 2007
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PROG
| (Other) sage: [floor(n/4) - 1 for n in xrange(4, 84)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2009]
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CROSSREFS
| Cf. A008615, A008621.
Zero followed by partial sums of A011765.
Partial sums: A130519. Other related sequences: A004526, A010872, A010873, A010874.
Third row of A180969.
Sequence in context: A091373 A197637 A008621 * A110655 A144075 A128929
Adjacent sequences: A002262 A002263 A002264 * A002266 A002267 A002268
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Clarified my formulas Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 01 2009
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