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%I #142 Dec 30 2023 10:56:11
%S 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,
%T 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,
%U 14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19
%N Nonnegative integers repeated 4 times.
%C 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
%C Complement of A010873, since A010873(n)+4*a(n)=n. - _Hieronymus Fischer_, Jun 01 2007
%C 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
%C 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
%C 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
%C Number of partitions of n into 4 parts whose smallest 3 parts are equal. - _Wesley Ivan Hurt_, Jan 17 2021
%D V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.
%H Todd Silvestri, <a href="/A002265/b002265.txt">Table of n, a(n) for n = 0..999</a>
%H J. H. McClellan and T. W. Parks, <a href="http://dx.doi.org/10.1109/TAU.1972.1162342">Eigenvalue and Eigenvector Decomposition of the Discrete Fourier Transform</a>, IEEE Trans. Audio and Electroacoust., Vol. AU-20, No. 1, March 1972, pp. 66-74.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = floor(n/4), n>=0;
%F G.f.: (x^4)/((1-x)*(1-x^4)).
%F 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
%F a(n) = (1/4)*(n-(3-(-1)^n-2*(-1)^((2*n-1+(-1)^n)/4))/2). - _Hieronymus Fischer_, Jul 04 2007
%F 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
%F 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
%F a(n) = A180969(2,n). - _Adriano Caroli_, Nov 26 2010
%F a(n) = A173562(n)-A000290(n); a(n+2) = A035608(n)-A173562(n). - _Reinhard Zumkeller_, Feb 21 2010
%F a(n+1) = A140201(n) - A057353(n+1). - _Reinhard Zumkeller_, Feb 26 2011
%F a(n) = ceiling((n-3)/4), n >= 0. - _Wesley Ivan Hurt_, Jun 01 2013
%F a(n) = (2*n + (-1)^n + 2*sin(Pi*n/2) + 2*cos(Pi*n/2) - 3)/8. - _Todd Silvestri_, Oct 27 2014
%F E.g.f.: (x/4 - 3/8)*exp(x) + exp(-x)/8 + (sin(x)+cos(x))/4. - _Robert Israel_, Oct 30 2014
%F 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
%F a(n) = A004526(A004526(n)). - _Bruno Berselli_, Jul 01 2016
%F From _Guenther Schrack_, May 03 2019: (Start)
%F a(n) = (2*n - 3 + (-1)^n + 2*(-1)^(n*(n-1)/2))/8.
%F a(n) = a(n-4) + 1, a(k)=0 for k=0,1,2,3, for n > 3. (End)
%p A002265:=n->floor(n/4); seq(A002265(n), n=0..100); # _Wesley Ivan Hurt_, Dec 10 2013
%t Table[Floor[n/4], {n, 0, 100}] (* _Wesley Ivan Hurt_, Dec 10 2013 *)
%t Table[{n,n,n,n},{n,0,20}]//Flatten (* _Harvey P. Dale_, Aug 08 2020 *)
%o (Sage) [floor(n/4) for n in range(0,84)] # _Zerinvary Lajos_, Dec 02 2009
%o (PARI) a(n)=n\4 \\ _Charles R Greathouse IV_, Dec 10 2013
%o (Magma) [Floor(n/4): n in [0..80]]; // _Vincenzo Librandi_, Oct 28 2014
%o (Python)
%o def A002265(n): return n>>2 # _Chai Wah Wu_, Jul 27 2022
%Y Cf. A008615, A008621, A249356.
%Y Zero followed by partial sums of A011765.
%Y Partial sums: A130519. Other related sequences: A004526, A010872, A010873, A010874.
%Y Third row of A180969.
%K nonn,easy
%O 0,9
%A _N. J. A. Sloane_
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