A293317 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 5, 11, 21, 38, 66, 112, 187, 310, 509, 832, 1355, 2202, 3573, 5792, 9383, 15194, 24598, 39814, 64435, 104273, 168733, 273032, 441792, 714852, 1156673, 1871555, 3028259, 4899846, 7928138, 12828018, 20756191, 33584245, 54340474, 87924758, 142265272

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

Links

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

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(0) -1 = 5;
a(3) = a(2) + a(1) + b(1) - 1 = 11.
Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15,...)

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 - 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}]  (* A293317 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293076.

Sequence in context: A093328 A045515 A298342 * A230097 A146787 A147243

Adjacent sequences: A293314 A293315 A293316 * A293318 A293319 A293320

Keyword

nonn,easy

Author

Clark Kimberling, Oct 28 2017

Status

approved