login
Second coordinate of (5,8)-Lagrange pair for n.
3

%I #12 Dec 29 2020 10:52:04

%S 2,-1,1,-2,0,2,-1,1,3,0,2,4,1,3,0,2,-1,1,3,0,2,4,1,3,5,2,4,1,3,0,2,4,

%T 1,3,5,2,4,6,3,5,2,4,1,3,5,2,4,6,3,5,7,4,6,3,5,2,4,6,3,5,7,4,6,8,5,7,

%U 4,6,3,5,7,4,6,8,5,7,9,6,8,5,7,4,6,8,5,7,9,6,8,10,7,9,6,8,5

%N Second coordinate of (5,8)-Lagrange pair for n.

%C See A194508.

%F From _Chai Wah Wu_, Jan 24 2020: (Start)

%F a(n) = a(n-1) + a(n-13) - a(n-14) for n > 14.

%F G.f.: x*(-3*x^12 + 2*x^11 + 2*x^10 - 3*x^9 + 2*x^8 + 2*x^7 - 3*x^6 + 2*x^5 + 2*x^4 - 3*x^3 + 2*x^2 - 3*x + 2)/(x^14 - x^13 - x + 1). (End)

%F a(n) = 7*n - 5*floor((9*n + 3)/13) - 5*floor((9*n + 8)/13). - _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)..-3..2.-1..4..1.-2..3..0.-3..2..-1..-4...1

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

%t c = 5; d = 8;

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

%t y1 = {2, -1, 1, -2, 0, 2, -1, 1, 3, 0, 2, 4, 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}] (* A194528 *)

%t Table[y[n], {n, 1, 100}] (* A194529 *)

%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, 40}]]

%Y Cf. A194508, A194528.

%K sign

%O 1,1

%A _Clark Kimberling_, Aug 28 2011