A294533 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3.

1, 2, 7, 14, 27, 48, 84, 142, 237, 391, 641, 1046, 1703, 2766, 4487, 7272, 11779, 19072, 30873, 49968, 80865, 130858, 211749, 342634, 554412, 897076, 1451519, 2348627, 3800179, 6148840, 9949054, 16097930, 26047021, 42144989, 68192049, 110337078, 178529168

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622)..

Links

Table of n, a(n) for n=0..36.

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(1) + a(0) + b(0) + 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 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] + 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}]  (* A294533 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A227016 A268347 A210728 * A294541 A294564 A068040

Adjacent sequences: A294530 A294531 A294532 * A294534 A294535 A294536

Keyword

nonn,easy

Author

Clark Kimberling, Nov 03 2017

Status

approved