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A194523 Second coordinate of (4,5)-Lagrange pair for n.

%I A194523 #15 Dec 30 2020 03:03:13
%S A194523 1,2,1,0,1,2,-1,0,1,2,3,2,1,2,3,0,1,2,3,4,3,2,3,4,1,2,3,4,5,4,3,4,5,2,
%T A194523 3,4,5,6,5,4,5,6,3,4,5,6,7,6,5,6,7,4,5,6,7,8,7,6,7,8,5,6,7,8,9,8,7,8,
%U A194523 9,6,7,8,9,10,9,8,9,10,7,8,9,10,11,10,9,10,11,8,9,10,11,12,11,10,11,12
%N A194523 Second coordinate of (4,5)-Lagrange pair for n.
%C A194523 See A194508.
%H A194523 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,1,-1).
%F A194523 From _Chai Wah Wu_, Jan 21 2020: (Start)
%F A194523 a(n) = a(n-1) + a(n-9) - a(n-10) for n > 10.
%F A194523 G.f.: x*(x^8 + x^7 - 3*x^6 + x^5 + x^4 - x^3 - x^2 + x + 1)/(x^10 - x^9 - x + 1). (End)
%F A194523 a(n) = n - 4*floor((n + 2)/9) - 2*floor((n + 5)/9) - 2*floor((n + 6)/9). - _Ridouane Oudra_, Dec 29 2020
%e A194523 This table shows (x(n),y(n)) for 1<=n<=13:
%e A194523 n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
%e A194523 x(n).. -1.-2..2..1..0.-1..3..2..1..0..-1...3...2
%e A194523 y(n)... 1..2..1..0..1..2.-1..0..1..2...3...2...1
%t A194523 Remove["Global`*"];
%t A194523 c = 4; d = 5;
%t A194523 x1 = {-1, -2, 2, 1, 0, -1, 3, 2, 1}; y1 = {1, 2, 1, 0, 1, 2, -1, 0, 1};
%t A194523 x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
%t A194523 y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
%t A194523 Table[x[n], {n, 1, 100}]  (* A194522 *)
%t A194523 Table[y[n], {n, 1, 100}]  (* A194523 *)
%t A194523 r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
%t A194523 TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
%Y A194523 Cf. A194508, A194522.
%K A194523 sign
%O A194523 1,2
%A A194523 _Clark Kimberling_, Aug 28 2011