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.

The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.33..., 1.49...

LINKS

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

EXAMPLE

a(0) = 3, a(1) = 4, b(0) = 1

a(2) = 2*a(0) - b(0) + 2 = 7

Complement: (b(n)) = (1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, ... )

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

a[0] = 3; a[1] = 4; b[0] = 1;

a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + n;

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 18}] (* A295069 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 19 2017

STATUS

approved