A300075 -id:A300075 - cp's OEIS frontend

A300076 A sequence based on the period 6 sequence A300075.

0, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 13, 13, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 19, 19, 20, 20, 20, 21, 22, 22, 23, 23, 23, 24, 25, 25, 26, 26, 26, 27, 28, 28, 29, 29, 29, 30, 31, 31, 32, 32, 32, 33, 34, 34, 35, 35, 35, 36, 37, 37, 38, 38, 38, 39, 40, 40, 41, 41, 41, 42, 43, 43, 44, 44, 44

Offset: 0

Wolfdieter Lang, Mar 03 2018

If 1 is added to each entry and the offset is set to 1 then the resulting sequence can be used to obtain integers in the quadratic number field Q(sqrt(3)) for the two components of the vertices V0_{-k}, as well as V3_{-k}, for k >= 1, of a k-family of ascending regular hexagons. Their centers 0{-k} form part of a discrete hexagon spiral.

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

Cf. A300068, A174257, A300075, A300293.

a(n) = A300075(n) + 3*floor(n/6), n >= 0.
a(n) = A300293(n-1) + 1, n >= 1.
G.f.: x*(1 + x^2 + x^5)/((1 - x^6)*(1 - x))  =  G(x) +  3*x^6/((1-x)*(1-x^6)), with the g.f. G(x) of A300075.

A300290 Period 6: repeat [0, 1, 2, 2, 3, 3].

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3, 0, 1, 2, 2, 3, 3

Offset: 0

Wolfdieter Lang, Mar 03 2018

Underlying A300068(n+2), n >= 0.

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

Cf. A300067, A300069, A300068, A300075.

a(n) = n (mod 6) - floor((n (mod 6))/3) - floor((n (mod 6))/5), n >= 0.
G.f.: x*(1 + x*(2 + 3*x^2)*(1 + x))/(1 - x^6).
a(n) = (11 - 5*cos(n*Pi/3) - 5*cos(2*n*Pi/3) - cos(n*Pi) - 3*sqrt(3)*sin(n*Pi/3) - sqrt(3)*sin(2*n*Pi/3))/6. - Wesley Ivan Hurt, Oct 04 2018

cp's OEIS Frontend

A300076 A sequence based on the period 6 sequence A300075.

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