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A047237
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Numbers that are congruent to {0, 1, 2, 4} mod 6.
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6
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0, 1, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 97
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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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Starting (1, 2, 4, 6, ...) = partial sums of (1, 1, 2, 2, 1, 1, 2, 2, ...). - Gary W. Adamson, Jun 19 2008
G.f.: x^2*(1+2*x^2) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (6*n - 8 + i^(1-n) - i^(1+n))/4 where i=sqrt(-1).
a(n) = (6*n - 8 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=2, a(4)=4, for n > 4.
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/3 + log(3)/4. - Amiram Eldar, Dec 16 2021
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MAPLE
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MATHEMATICA
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LinearRecurrence[{2, -2, 2, -1}, {0, 1, 2, 4}, 120] (* Harvey P. Dale, Jan 21 2018 *)
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PROG
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(Magma) [n : n in [0..110] | n mod 6 in [0, 1, 2, 4]]; // G. C. Greubel, Feb 16 2019
(PARI) my(x='x+O('x^70)); concat([0], Vec(x^2*(1+2*x^2)/((1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019
(Sage) a=(x^2*(1+2*x^2)/((1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019
(GAP) Filtered([0..100], n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 4); # Muniru A Asiru, Feb 19 2019
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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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EXTENSIONS
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STATUS
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approved
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