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A300068
A sequence based on the period 6 sequence A300067.
7
0, 0, 0, 1, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 20, 20, 21, 21, 21, 22, 23, 23, 24, 24, 24, 25, 26, 26, 27, 27, 27, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 33, 33, 34, 35, 35, 36, 36, 36, 37, 38, 38, 39, 39, 39, 40, 41, 41
OFFSET
0,5
COMMENTS
From Wolfdieter Lang, Mar 30 2018: (Start)
a(k) + 2 =: s(k) is used to obtain for 2^s(k)*vec v_{-k} integer components in the quadratic number field Q(sqrt(3)), where vec v_{-k} = vec(O_{-(k+1)}, O_{-k})) with the centers O_{-k}, k >= 0, for a k-family of regular hexagons H_{-k} forming part of a discrete spiral. See the linked paper, Lemma 4 and Table 7.
a(k+2) =: v0(k), k >= 0, based on the sequence A300290, is used to obtain for
2^(v0(k))*V_{-k}(0) as well as 2^(v0(k))*V_{-k}(3) integer coordinates in the quadratic number field Q(sqrt(3)), where V_{-k}(j), j = 0..5, are the vertices of the regular hexagon H_{-k}, of the above mentioned k-family. See the linked paper, Lemma 6 and Table 8.
a(k+1) + 1 =: v1(k), k >= 1, is used to obtain for 2^(v1(k))*V_{-k}(1) as well as 2^(v1(k))*V_{-k}(4) integer coordinates in the quadratic number field Q(sqrt(3)), with vertices V_{-k}(j) of H_{-k}. See the linked paper, Lemma 6 and Table 9.
(End)
FORMULA
a(n) = A300067(n) + 3*floor(n/6), n >= 0.
G.f.: x^3*(1 + x + x^3)/((1 - x^6)*(1 - x)).
a(n+2) = A300290(n) + 3*floor(n/6), n >= 0.
a(n) = (6*n - 5 + cos(n*Pi) + 4*cos((n+1)*Pi/3) - 4*cos(2*(n+1)*Pi/3))/12. - Wesley Ivan Hurt, Oct 04 2018
MAPLE
ListTools:-PartialSums(map(op, [[0], [0, 0, 1, 1, 0, 1]$30])); # Robert Israel, Mar 25 2018
MATHEMATICA
CoefficientList[Series[x^3*(1 + x + x^3)/((1 - x^6) (1 - x)), {x, 0, 102}], x] (* or *)
MapIndexed[#1 + 3 Floor[(First[#2] - 1)/6] &, PadRight[{}, 102, {0, 0, 0, 1, 2, 2}]] (* Michael De Vlieger, Feb 25 2018 *)
PROG
(PARI) a300067(n) = my(v=[0, 0, 1, 2, 2]); v[if(n%6==0, 1, n%6)]
a(n) = a300067(n) + 3*floor(n/6) \\ Felix Fröhlich, Feb 24 2018
(PARI) concat([0, 0, 0], Vec(x^3*(1 + x + x^3)/((1 - x^6)*(1 - x)) + O(x^40))) \\ Felix Fröhlich, Feb 24 2018
CROSSREFS
Sequence in context: A140473 A115384 A131411 * A194202 A261224 A125059
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Feb 24 2018
STATUS
approved