A300293 A sequence based on the period 6 sequence A151899.

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

Offset

0,6

Comments

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

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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) = A151899(n) + 3*floor(n/6), n >= 0.

a(n) = A300076(n+1) - 1.

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.

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 0, 1, 1, 1, 2, 3}, 100] (* Harvey P. Dale, Dec 29 2024 *)

Prog

(PARI) a151899(n) = [0, 0, 1, 1, 1, 2][n%6+1]
a(n) = a151899(n) + 3*floor(n/6) \\ Felix Fröhlich, Mar 30 2018

Crossrefs

Cf. A174257, A300068, A300076, A151899.

Sequence in context: A110862 A104257 A048182 * A316388 A029107 A209727

Adjacent sequences: A300290 A300291 A300292 * A300294 A300295 A300296

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 05 2018

Status

approved