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

0,2

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

Cf. A001622 (golden ratio), A293765.

Sequence in context: A342174 A159791 A078723 * A296556 A141187 A107138

Adjacent sequences: A294365 A294366 A294367 * A294369 A294370 A294371

Keyword

nonn,easy

Author

Clark Kimberling, Oct 29 2017

Extensions

Example clarified by Robert Israel, Oct 29 2017

Status

approved