Search: id:A002265
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%I A002265
%S A002265 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,
%T A002265 8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,
%U A002265 14,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19
%N A002265 Integers repeated 4 times.
%C A002265 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
%C A002265 After initial terms, same as [n/2] - [n/4]. - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jan 19 2007
%C A002265 Complement of A010873, since A010873(n)+4*a(n)=n. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
%D A002265 V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, 
               Bristol 1986, p. 61.
%D A002265 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.
%H A002265 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A002265 a(n) = floor(n/4), n>=0;
%F A002265 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
%F A002265 G.f.: (x^4)/((1-x)*(1-x^4))
%F A002265 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
%F A002265 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
%F A002265 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
%F A002265 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
%p A002265 seq(seq(seq(k,i=2..3),j=2..3),k=0..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 29 2007
%p A002265 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
%o A002265 (Other) sage: [floor(n/4) - 1 for n in xrange(4,84)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2009]
%Y A002265 Cf. A008621.
%Y A002265 Zero followed by partial sums of A011765.
%Y A002265 Cf. A008615.
%Y A002265 Partial sums: A130519. Other related sequences: A004526, A010872, A010873, 
               A010874.
%Y A002265 Sequence in context: A056172 A091373 A008621 this_sequence A110655 A144075 
               A128929
%Y A002265 Adjacent sequences: A002262 A002263 A002264 this_sequence A002266 A002267 
               A002268
%K A002265 nonn,easy,new
%O A002265 0,9
%A A002265 N. J. A. Sloane (njas(AT)research.att.com).
%E A002265 Clarified my formulas Mohammad K. Azarian (azarian(AT)evansville.edu), 
               Aug 01 2009

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