0,9

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

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.

Index entries for sequences related to linear recurrences with constant coefficients

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 (ppl(AT)spl.at), 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 (ppl(AT)spl.at), 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

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 (ppl(AT)spl.at), Jun 20 2007

(Other) sage: [floor(n/4) - 1 for n in xrange(4, 84)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2009]

Cf. A008621.

Zero followed by partial sums of A011765.

Cf. A008615.

Partial sums: A130519. Other related sequences: A004526, A010872, A010873, A010874.

Sequence in context: A056172 A091373 A008621 this_sequence A110655 A144075 A128929

Adjacent sequences: A002262 A002263 A002264 this_sequence A002266 A002267 A002268

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

Clarified my formulas Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 01 2009