A295058 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.

3, 5, 8, 12, 18, 29, 49, 88, 165, 317, 620, 1225, 2434, 4851, 9683, 19346, 38671, 77320, 154617, 309210, 618395, 1236764, 2473501, 4946974, 9893918, 19787805, 39575578, 79151123, 158302212, 316604389, 633208742

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..30.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Example

a(0) = 3, a(1) = 5, b(0) = 1
b(1) = 2 (least "new number")
a(2) = 2*a(1) - b(1) = 8
Complement: (b(n)) = (1, 2, 4, 6, 7, 9, 10, 11, 13, 14, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] - b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A295058 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A295053.

Sequence in context: A020745 A232896 A227635 * A004398 A286311 A256057

Adjacent sequences: A295055 A295056 A295057 * A295059 A295060 A295061

Keyword

nonn,easy

Author

Clark Kimberling, Nov 18 2017

Status

approved