A295058 - OEIS

%I #4 Nov 18 2017 20:54:50

%S 3,5,8,12,18,29,49,88,165,317,620,1225,2434,4851,9683,19346,38671,

%T 77320,154617,309210,618395,1236764,2473501,4946974,9893918,19787805,

%U 39575578,79151123,158302212,316604389,633208742

%N Solution of the complementary equation a(n) = 2*a(n-1) - b(n-1), where a(0) = 3, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%e a(0) = 3, a(1) = 5, b(0) = 1

%e b(1) = 2 (least "new number")

%e a(2) = 2*a(1) - b(1) = 8

%e Complement: (b(n)) = (1, 2, 4, 6, 7, 9, 10, 11, 13, 14, 15, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 3; a[1] = 5; b[0] = 1;

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

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 18}]  (* A295058 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A295053.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Nov 18 2017