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A047273
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Numbers that are congruent to {0, 1, 3, 5} mod 6.
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11
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0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: x*(1+x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^2)*(1-x^3)/((1-x)^3*(1-x^4)).
a(n) = n + A004524(n+1) = -a(-n) for all n in Z.
Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
a(n) = 2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, with offset 0..a(0)=0. - Gary Detlefs, Mar 19 2010
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.
a(n) = (6*n-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)
Euler transform of length 4 sequence [3, -1, -1, 1]. - Michael Somos, Jun 24 2017
Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/2. - Amiram Eldar, Dec 16 2021
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MAPLE
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seq(2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, n = 0..69); # Gary Detlefs, Mar 19 2010
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MATHEMATICA
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LinearRecurrence[{2, -2, 2, -1}, {0, 1, 3, 5}, 80] (* Harvey P. Dale, Jan 04 2015 *)
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PROG
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(PARI) a(n)=n+(n+1)\4+(n+2)\4
(Sage) [(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # Zerinvary Lajos, Mar 09 2009
(Haskell)
a047273 n = a047273_list !! (n-1)
a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list
(Magma) [[(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 20 2016
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CROSSREFS
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See A301729 for an essentially identical sequence.
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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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