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A300293 A sequence based on the period 6 sequence A151899.

%I A300293 #37 Jun 19 2025 22:27:05
%S A300293 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,
%T A300293 14,15,15,16,16,16,17,18,18,19,19,19,20,21,21,22,22,22,23,24,24,25,25,
%U A300293 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
%N A300293 A sequence based on the period 6 sequence A151899.
%C A300293 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
%H A300293 Harvey P. Dale, <a href="/A300293/b300293.txt">Table of n, a(n) for n = 0..1000</a>
%H A300293 Wolfdieter Lang, <a href="/A300068/a300068_1.pdf">On a Conformal Mapping of Regular Hexagons and the Spiral of its Centers</a>
%H A300293 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A300293 a(n) = A151899(n) + 3*floor(n/6), n >= 0.
%F A300293 a(n) = A300076(n+1) - 1.
%F A300293 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.
%F A300293 a(n) = a(n-1) + a(n-6) - a(n-7). - _Wesley Ivan Hurt_, Jun 19 2025
%t A300293 LinearRecurrence[{1,0,0,0,0,1,-1},{0,0,1,1,1,2,3},100] (* _Harvey P. Dale_, Dec 29 2024 *)
%o A300293 (PARI) a151899(n) = [0, 0, 1, 1, 1, 2][n%6+1]
%o A300293 a(n) = a151899(n) + 3*floor(n/6) \\ _Felix Fröhlich_, Mar 30 2018
%Y A300293 Cf. A174257, A300068, A300076, A151899.
%K A300293 nonn,easy
%O A300293 0,6
%A A300293 _Wolfdieter Lang_, Mar 05 2018