OFFSET
0,2
COMMENTS
The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.
LINKS
Robert Israel, Table of n, a(n) for n = 0..4775
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 3 = 11;
b(2) is the first positive integer not already seen, namely 5.
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)
MAPLE
A[0]:= 1: B[0]:= 2:
A[1]:= 3: B[1]:= 4:
Av:= {$5..200}:
for n from 2 to 100 do
A[n]:= A[n-1]+A[n-2]+B[n-1]+n+1;
B[n]:= min(Av minus {A[n]});
Av:= Av minus {A[n], B[n]};
od:
seq(A[i], i=0..100); # Robert Israel, Oct 29 2017
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n + 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}] (* A294368 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 29 2017
EXTENSIONS
Example clarified by Robert Israel, Oct 29 2017
STATUS
approved