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A047222
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Numbers that are congruent to {0, 2, 3} mod 5.
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29
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0, 2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 107
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OFFSET
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1,2
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COMMENTS
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Row sum of a triangle where the top value is 2 and every elementary triangle or triple is required to have the values 1,2,2 (see link below). Compare with A008854 where the triple contains 1,2,2 with 1 at the top. - Craig Knecht, Oct 18 2015
Also, numbers k such that k*(k^2+1)/5 is a nonnegative integer. - Bruno Berselli, Jan 16 2016
Conjecture: Apart from 0, the sequence gives the values for c/6, such that an infinite number of primes, p, result in both p^2-c and p^2+c being positive primes, except when c is a square. When c is square solutions exist for c (both within and outside of the a(n) set), but occur at only a single prime p. See A274609. Other c values with only one prime providing a solution occur when p^2-c=3. See A274610. The only remaining c values with single p solutions are: c=2 (with p=3) and c=6 (with p=5). - Richard R. Forberg, Jun 26 2016
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LINKS
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FORMULA
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G.f.: x^2*(2 + x + 2*x^2)/((1 - x)^2*(1 + x + x^2)).
a(n) = A028738(n-2), 1 < n < 16. (End)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-15-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k-2, a(3k-1) = 5k-3, a(3k-2) = 5k-5. (End)
Sum_{n>=2} (-1)^n/a(n) = arccoth(3/sqrt(5))/sqrt(5) - log(2)/5. - Amiram Eldar, Dec 10 2021
a(n) = a(floor(n/2)) + a(1 + ceiling(n/2)) for n >= 4 with a(1) = 0, a(2) = 2 and a(3) = 3.
a(2*n) = a(n) + a(n+1); a(2*n+1) = a(n) + a(n+2). Cf. A008854 and A042965. (End)
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MAPLE
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MATHEMATICA
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Flatten[Table[5n + {0, 2, 3}, {n, 0, 19}]] (* Alonso del Arte, Nov 07 2013 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 3, 5}, 100] (* Vincenzo Librandi, Jun 15 2016 *)
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PROG
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(PARI) concat(0, Vec(x^2*(2+x+2*x^2)/((1-x)^2*(1+x+x^2)) + O(x^100))) \\ Altug Alkan, Oct 26 2015
(Magma) [n : n in [0..150] | n mod 5 in [0, 2, 3]]; // Wesley Ivan Hurt, Jun 14 2016
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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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