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A294544 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 10, 20, 39, 69, 119, 200, 333, 548, 897, 1462, 2377, 3858, 6255, 10134, 16411, 26569, 43005, 69600, 112632, 182260, 294921, 477211, 772163, 1249406, 2021602, 3271042, 5292679, 8563757, 13856473, 22420268, 36276780, 58697088, 94973909, 153671040 (list; graph; refs; listen; history; text; internal format)
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..35.

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(1) + 3 = 10.

Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...).

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 - 1] + 3;

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 40}]  (* A294544 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

Cf. A001622, A294532.

Sequence in context: A177150 A165551 A139592 * A285104 A120552 A090220

Adjacent sequences:  A294541 A294542 A294543 * A294545 A294546 A294547

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 04 2017

STATUS

approved

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)