OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3:
a(n) = a(n-1) + a(n-2) + b(n-2) A294532
a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533
a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534
a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535
a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536
a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538
a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540
a(n) = a(n-1) + a(n-2) + b(n-1) A294541
a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542
a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543
a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544
a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545
a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546
a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547
a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548
a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558
a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294559
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294560
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294561
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1 A294564
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 3 A294565
Conjecture: for every sequence listed here, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(0) + a(1) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294532 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 03 2017
STATUS
approved