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A300293
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A sequence based on the period 6 sequence A151899.
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1
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0, 0, 1, 1, 1, 2, 3, 3, 4, 4, 4, 5, 6, 6, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 21, 22, 22, 22, 23, 24, 24, 25, 25, 25, 26, 27, 27, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36, 36, 37, 37, 37, 38, 39, 39, 40, 40, 40, 41, 42, 42, 43, 43, 43, 44
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OFFSET
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0,6
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COMMENTS
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a(k-1) + 2 =: v2(k), k >= 1, is used to obtain for 2^(v2(k))*V_{-k}(2) as well as 2^(v2(k))*V_{-k}(5) integer coordinates in the quadratic number field Q(sqrt(3)), where V_{-k}(j), j = 0..5, are the vertices of a k-family of regular hexagons H_{-k} whose centers O_{-k} form part of a discrete spiral. See the linked paper, Lemma 6, eqs. (47) and (48), and the Table 19. - Wolfdieter Lang, Mar 30 2018
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LINKS
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FORMULA
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a(n) = A151899(n) + 3*floor(n/6), n >= 0.
G.f.: x^2*(1 + x^3 + x^4)/((1 - x^6)*(1 - x)) = G(x) + 3*x^6/((1-x)*(1-x^6)), with the g.f. G(x) of A151899.
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PROG
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(PARI) a151899(n) = [0, 0, 1, 1, 1, 2][n%6+1]
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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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