A294554 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 12, 25, 50, 90, 157, 266, 444, 733, 1203, 1965, 3199, 5197, 8431, 13665, 22135, 35841, 58019, 93905, 151971, 245925, 397948, 643928, 1041933, 1685920, 2727914, 4413897, 7141876, 11555840, 18697785, 30253696, 48951554, 79205325, 128156956, 207362360

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) + 2 = 12
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294554 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A092825 A135396 A031048 * A098707 A152811 A294552

Adjacent sequences: A294551 A294552 A294553 * A294555 A294556 A294557

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved