OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
FORMULA
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that
b(2) = 5 (least "new number")
a(2) = a(1) + a(0) + b(1)*b(0) = 15
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 32; u = Table[a[n], {n, 0, z}] (* A295367 *)
v = Table[b[n], {n, 0, 10}] (* complement *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 21 2017
STATUS
approved