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A047270
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Numbers that are congruent to {3, 5} mod 6.
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16
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3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 33, 35, 39, 41, 45, 47, 51, 53, 57, 59, 63, 65, 69, 71, 75, 77, 81, 83, 87, 89, 93, 95, 99, 101, 105, 107, 111, 113, 117, 119, 123, 125, 129, 131, 135, 137, 141, 143, 147, 149
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OFFSET
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1,1
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COMMENTS
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Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 10 ).
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LINKS
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FORMULA
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a(n) = sqrt(2)*sqrt((1-6*n)*(-1)^n + 18*n^2 - 6*n + 1)/2. - Paul Barry, May 11 2003
G.f.: (3+2*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0, with n > 3.
a(n) = (6*n - (-1)^n - 1)/2. (End)
a(n) = a(n-2) + 6 for n > 2.
a(n) = A109613(n-1) + 2*n for n > 0.
m-element moving averages: Sum_{k=1..m} a(n-m+k)/m = A016777(n-m/2) for m = 2, 4, 6, ... and n >= m. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f.: 1 + 3*x*exp(x) - cosh(x). - David Lovler, Aug 25 2022
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MATHEMATICA
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Select[Range@ 149, MemberQ[{3, 5}, Mod[#, 6]] &] (* or *)
Array[(6 # - (-1)^# - 1)/2 &, 50] (* or *)
Fold[Append[#1, 6 #2 - Last@ #1 - 4] &, {3}, Range[2, 50]] (* or *)
CoefficientList[Series[(3 + 2 x + x^2)/((1 + x) (1 - x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 12 2018 *)
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PROG
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(PARI) a(n) = (6*n - 1 - (-1)^n)/2 \\ David Lovler, Aug 25 2022
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CROSSREFS
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First differences: A105397(n) for n > 0.
Partial sums: A227017(n+1) for n > 0.
Elements of even index: A016969(n-1) for n > 0. (End)
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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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