A294551 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 11, 23, 46, 83, 145, 246, 411, 680, 1117, 1825, 2972, 4829, 7835, 12700, 20573, 33313, 53928, 87285, 141260, 228595, 369907, 598556, 968519, 1567133, 2535712, 4102907, 6638683, 10741656, 17380407, 28122133, 45502612, 73624819, 119127507, 192752404

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

Links

Table of n, a(n) for n=0..35.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number");
a(2) = a(1) + a(0) + b(1) + b(0) + 1 = 11.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...).

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294551 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A045387 A103255 A031385 * A179878 A126916 A090424

Adjacent sequences: A294548 A294549 A294550 * A294552 A294553 A294554

Keyword

nonn,easy

Author

Clark Kimberling, Nov 04 2017

Status

approved