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A194509 Second coordinate of (2,3)-Lagrange pair for n.

%I A194509 #31 Dec 15 2021 21:43:32
%S A194509 1,0,1,0,1,2,1,2,1,2,3,2,3,2,3,4,3,4,3,4,5,4,5,4,5,6,5,6,5,6,7,6,7,6,
%T A194509 7,8,7,8,7,8,9,8,9,8,9,10,9,10,9,10,11,10,11,10,11,12,11,12,11,12,13,
%U A194509 12,13,12,13,14,13,14,13,14,15,14,15,14,15,16,15,16,15,16,17,16
%N A194509 Second coordinate of (2,3)-Lagrange pair for n.
%C A194509 See A194508.
%H A194509 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A194509 From _Chai Wah Wu_, Jan 21 2020: (Start)
%F A194509 a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
%F A194509 G.f.: x*(x^4 - x^3 + x^2 - x + 1)/(x^6 - x^5 - x + 1). (End)
%F A194509 a(n) = n - 2*floor((2*n + 2)/5). - _Ridouane Oudra_, Dec 25 2020
%F A194509 a(n) = a(n-1) + (-1)^((n-1) mod 5) for n > 1. - _Alexander Van Plantinga_, Dec 14 2021
%e A194509 This table shows (x(n),y(n)) for 1 <= n <= 13:
%e A194509 n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
%e A194509 x(n).. -1..1..0..2..1..0..2..1..3..2...1...3...2
%e A194509 y(n)... 1..0..1..0..1..2..1..2..1..2...3...2...3
%t A194509 c = 2; d = 3;
%t A194509 x1 = {-1, 1, 0, 2, 1}; y1 = {1, 0, 1, 0, 1};
%t A194509 x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
%t A194509 y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
%t A194509 Table[x[n], {n, 1, 100}] (* A194508 *)
%t A194509 Table[y[n], {n, 1, 100}] (* A194509 *)
%t A194509 r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
%t A194509 TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
%Y A194509 Cf. A194508.
%K A194509 nonn,easy
%O A194509 1,6
%A A194509 _Clark Kimberling_, Aug 27 2011