A293357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n +1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 9, 20, 39, 71, 124, 211, 354, 586, 963, 1574, 2564, 4167, 6762, 10962, 17759, 28758, 46557, 75357, 121958, 197361, 319367, 516778, 836197, 1353029, 2189282, 3542369, 5731711, 9274142, 15005917, 24280125, 39286110, 63566305, 102852487, 166418866

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..35.

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + a(0) + b(0) + 3 = 9;
a(3) = a(2) + a(1) + b(1) + 4 = 20.
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14,...)

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] + n + 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}]  (* A293357 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293076.

Sequence in context: A225385 A037257 A145068 * A202349 A192951 A027114

Adjacent sequences: A293354 A293355 A293356 * A293358 A293359 A293360

Keyword

nonn,easy

Author

Clark Kimberling, Oct 28 2017

Status

approved