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A008233
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a(n) = floor(n/4)*floor((n+1)/4)*floor((n+2)/4)*floor((n+3)/4).
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13
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0, 0, 0, 0, 1, 2, 4, 8, 16, 24, 36, 54, 81, 108, 144, 192, 256, 320, 400, 500, 625, 750, 900, 1080, 1296, 1512, 1764, 2058, 2401, 2744, 3136, 3584, 4096, 4608, 5184, 5832, 6561, 7290, 8100, 9000, 10000, 11000, 12100, 13310, 14641, 15972, 17424, 19008, 20736
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OFFSET
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0,6
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COMMENTS
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a(n) is the maximal product of four nonnegative integers whose sum is n. - Andres Cicuttin, Sep 26 2018
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).
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FORMULA
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Let b(n) = A002620(n), the quarter-squares. Then this sequence is b(0)*b(0), b(0)*b(1), b(1)*b(1), b(1)*b(2), b(2)*b(2), b(2)*b(3), ...
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14).
G.f.: -x^4*(1+x^6+x^2+2*x^3+x^4) / ( (1+x)^3*(x^2+1)^3*(x-1)^5 ). (End)
Sum_{n>=4} 1/a(n) = 1 + zeta(4). - Amiram Eldar, Jan 10 2023
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MAPLE
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MATHEMATICA
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Table[Floor[n/4]*Floor[(n + 1)/4]*Floor[(n + 2)/4]*Floor[(n + 3)/4], {n, 0, 50}] (* Stefan Steinerberger, Apr 03 2006 *)
Table[Times@@Floor[Range[n, n+3]/4], {n, 0, 50}] (* Harvey P. Dale, Mar 30 2019 *)
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PROG
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(Haskell)
a008233 n = product $ map (`div` 4) [n..n+3]
(Magma) [Floor(n/4)*Floor((n+1)/4)*Floor((n+2)/4)*Floor((n+3)/4): n in [0..50]]; // Vincenzo Librandi, Jun 09 2011
(PARI) a(n) = prod(i=0, 3, (n+i)\4); \\ Altug Alkan, Sep 27 2018
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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