A294368 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 11, 23, 45, 81, 141, 239, 399, 660, 1083, 1769, 2880, 4679, 7591, 12304, 19931, 32273, 52244, 84559, 136848, 221454, 358351, 579856, 938260, 1518171, 2456488, 3974718, 6431267, 10406048, 16837380, 27243495, 44080944, 71324510, 115405527, 186730112

Offset: 0

Author

Clark Kimberling, Oct 29 2017

Keywords

• nonn
• easy

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.

Examples

			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, ...)
		

Links

• Robert Israel, Table of n, a(n) for n = 0..4775
• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A001622 (golden ratio), A293765.

Programs

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}]

Extensions

Example clarified by Robert Israel, Oct 29 2017