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A194516 First coordinate of (3,5)-Lagrange pair for n.

%I A194516 #17 Nov 29 2024 18:45:01
%S A194516 2,-1,1,3,0,2,-1,1,3,0,2,4,1,3,0,2,4,1,3,5,2,4,1,3,5,2,4,6,3,5,2,4,6,
%T A194516 3,5,7,4,6,3,5,7,4,6,8,5,7,4,6,8,5,7,9,6,8,5,7,9,6,8,10,7,9,6,8,10,7,
%U A194516 9,11,8,10,7,9,11,8,10,12,9,11,8,10,12,9,11,13,10,12,9,11,13,10,12,14,11,13,10
%N A194516 First coordinate of (3,5)-Lagrange pair for n.
%C A194516 See A194508.
%H A194516 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,1,-1).
%F A194516 From _Chai Wah Wu_, Jan 21 2020: (Start)
%F A194516 a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
%F A194516 G.f.: x*(2*x^7 - 3*x^6 + 2*x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 2)/(x^9 - x^8 - x + 1). (End)
%F A194516 a(n) = 2*n - 5*floor((3*n + 3)/8). - _Ridouane Oudra_, Dec 29 2020
%e A194516 This table shows (x(n),y(n)) for 1<=n<=13:
%e A194516 n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
%e A194516 x(n)... 2.-1..1..3..0..2.-1..1..3..0...2...4...1
%e A194516 y(n).. -1..1..0.-1..1..0..2..1..0..2...1...0...2
%t A194516 c = 3; d = 5;
%t A194516 x1 = {2, -1, 1, 3, 0, 2, -1, 1}; y1 = {-1, 1, 0, -1, 1, 0, 2, 1};
%t A194516 x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
%t A194516 y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
%t A194516 Table[x[n], {n, 1, 100}]  (* A194516 *)
%t A194516 Table[y[n], {n, 1, 100}]  (* A194517 *)
%t A194516 r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
%t A194516 TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
%t A194516 LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{2,-1,1,3,0,2,-1,1,3},100] (* _Harvey P. Dale_, Nov 29 2024 *)
%Y A194516 Cf. A194508, A194517.
%K A194516 sign
%O A194516 1,1
%A A194516 _Clark Kimberling_, Aug 28 2011