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A047220
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Numbers that are congruent to {0, 1, 3} mod 5.
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24
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0, 1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 85, 86, 88, 90, 91, 93, 95, 96, 98, 100, 101, 103, 105, 106
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OFFSET
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1,3
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COMMENTS
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Also numbers k such that k*(k+2)*(k+4) is divisible by 5. - Bruno Berselli, Dec 28 2017
Maximum sum of degeneracies over all decompositions of the complete graph of order n into four factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
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LINKS
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FORMULA
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a(n) = 2*n - floor(n/3) - (n^2 mod 3), with offset 0. - Gary Detlefs, Mar 19 2010
G.f.: x^2*(1 + 2*x + 2*x^2)/(1 - x)^2/(1 + x + x^2). - Colin Barker, Feb 17 2012
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 5*n/3 - 2 + 2*sin(2*n*Pi/3)/(3*sqrt(3)).
a(3*k) = 5*k-2, a(3*k-1) = 5*k-4, a(3*k-2) = 5*k-5. (End)
E.g.f.: 2 + (5*x - 6)*exp(x)/3 + 2*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2))/(3*sqrt(3)). - Ilya Gutkovskiy, Jun 14 2016
Sum_{n>=2} (-1)^n/a(n) = sqrt(1-2/sqrt(5))*Pi/5 + 2*log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 16 2023
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MAPLE
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seq(2*n-floor(n/3)-(n^2 mod 3), n=0..55); # Gary Detlefs, Mar 19 2010
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MATHEMATICA
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PROG
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(Magma) I:=[0, 1, 3, 5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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