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 A002265 Nonnegative integers repeated 4 times. 100
 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; text; internal format)
 OFFSET 0,9 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, Jan 21 2005 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 Number of partitions of n into 4 parts whose smallest 3 parts are equal. - Wesley Ivan Hurt, Jan 17 2021 REFERENCES V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61. LINKS Todd Silvestri, Table of n, a(n) for n = 0..999 J. H. McClellan and 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). FORMULA a(n) = floor(n/4), n>=0; a(n) = ( (Sum_{k=0..n} (k+1)*cos(Pi*(n-k)/2)) + 1/4*(2*cos(n*Pi/2)+1+(-1)^n) )/2 - 1. - Paolo P. Lava, Oct 09 2006 [corrected by Kevin Ryde, Sep 08 2022] 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 and Aug 01 2009 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 From Guenther Schrack, May 03 2019: (Start) a(n) = (2*n - 3 + (-1)^n + 2*(-1)^(n*(n-1)/2))/8. a(n) = a(n-4) + 1, a(k)=0 for k=0,1,2,3, for n > 3. (End) MAPLE A002265:=n->floor(n/4); seq(A002265(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013 MATHEMATICA Table[Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *) Table[{n, n, n, n}, {n, 0, 20}]//Flatten (* Harvey P. Dale, Aug 08 2020 *) PROG (Sage) [floor(n/4) for n in range(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 (Python) def A002265(n): return n>>2 # Chai Wah Wu, Jul 27 2022 CROSSREFS 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. Sequence in context: A347697 A300763 A359276 * A242601 A110655 A008621 Adjacent sequences: A002262 A002263 A002264 * A002266 A002267 A002268 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified June 1 17:57 EDT 2023. Contains 363076 sequences. (Running on oeis4.)