A305746 - OEIS

%I #8 Jun 12 2018 10:21:22

%S 1,2,3,12,30,66,130,241,429,742,1258,2103,3481,5722,9360,15259,24817,

%T 40296,65356,105919,171567,277804,449716,727893,1178011,1906337,

%U 3084813,4991648,8076993,13069208,21146804,34216652,55364134,89581503,144946394,234528695

%N Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n), where a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7. See Comments.

%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values; a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622).

%H Clark Kimberling, <a href="/A305746/b305746.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 1, a(1) = 2, a(2) = 3, b(0)= 4, b(1) = 5, b(2) = 6; b(3) = 7, and a(3) = 2*3 - 1 + 7 = 12.

%t a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6; b[3] = 7;

%t a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + b[n];

%t j = 1; While[j < 12, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t Table[a[n], {n, 0, 60}]  (* A305746 *)

%Y Cf. A001622, A296245.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 10 2018