A300068 A sequence based on the period 6 sequence A300067.

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)

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Wolfdieter Lang, On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

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

Cf. A300067, A300290.

Sequence in context: A140473 A115384 A131411 * A194202 A261224 A125059

Adjacent sequences: A300065 A300066 A300067 * A300069 A300070 A300071

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 24 2018

Status

approved