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Second coordinate of (4,7)-Lagrange pair for n.
3

%I #15 Dec 29 2020 10:21:04

%S -1,-2,1,0,-1,2,1,0,-1,2,1,0,-1,2,1,0,3,2,1,0,3,2,1,0,3,2,1,4,3,2,1,4,

%T 3,2,1,4,3,2,5,4,3,2,5,4,3,2,5,4,3,6,5,4,3,6,5,4,3,6,5,4,7,6,5,4,7,6,

%U 5,4,7,6,5,8,7,6,5,8,7,6,5,8,7,6,9,8,7,6,9,8,7,6,9,8,7

%N Second coordinate of (4,7)-Lagrange pair for n.

%C See A194508.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

%F From _Chai Wah Wu_, Jan 21 2020: (Start)

%F a(n) = a(n-1) + a(n-11) - a(n-12) for n > 12.

%F G.f.: x*(-x^10 + 3*x^9 - x^8 - x^7 - x^6 + 3*x^5 - x^4 - x^3 + 3*x^2 - x - 1)/(x^12 - x^11 - x + 1). (End)

%F a(n) = 3*n - 4*floor((8*n + 6)/11). - _Ridouane Oudra_, Dec 29 2020

%e This table shows (x(n),y(n)) for 1<=n<=13:

%e n..... 1..2..3..4..5..6..7..8..9..10..11..12..13

%e x(n).. 2..4.-1..1..3.-2..0..2..4.-1...1...3...5

%e y(n). -1.-2..1..0.-1..2..1..0.-1..2...1...0..-1

%t c = 4; d = 7;

%t x1 = {2, 4, -1, 1, 3, -2, 0, 2, 4, -1, 1};

%t y1 = {-1, -2, 1, 0, -1, 2, 1, 0, -1, 2, 1};

%t x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]

%t y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]

%t Table[x[n], {n, 1, 100}] (* A194524 *)

%t Table[y[n], {n, 1, 100}] (* A194525 *)

%t r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]

%t TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

%Y Cf. A194508, A194524.

%K sign

%O 1,2

%A _Clark Kimberling_, Aug 28 2011