A294564 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 14, 27, 50, 86, 146, 243, 401, 657, 1074, 1747, 2838, 4603, 7460, 12083, 19564, 31669, 51256, 82949, 134230, 217205, 351464, 568698, 920192, 1488921, 2409145, 3898099, 6307278, 10205412, 16512726, 26718175, 43230939, 69949153, 113180132, 183129326

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) + 2*b(1) - b(0) - 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 10, 11, 12, 13, 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] + 2 b[n - 1] - 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}]  (* A294564 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622, A294532.

Sequence in context: A210728 A294533 A294541 * A068040 A200084 A184704

Adjacent sequences: A294561 A294562 A294563 * A294565 A294566 A294567

Keyword

nonn,easy

Author

Clark Kimberling, Nov 15 2017

Status

approved