A294555 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 13, 27, 54, 97, 169, 286, 477, 787, 1290, 2106, 3428, 5568, 9032, 14638, 23710, 38390, 62144, 100580, 162772, 263402, 426226, 689682, 1115965, 1805707, 2921734, 4727505, 7649305, 12376878, 20026253, 32403203, 52429530, 84832809, 137262417, 222095306

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) + 3 = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 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] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294555 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A274869 A018400 A091052 * A031090 A358296 A294556

Adjacent sequences: A294552 A294553 A294554 * A294556 A294557 A294558

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved