A293057 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 17, 32, 57, 98, 166, 276, 455, 745, 1215, 1976, 3208, 5202, 8430, 13653, 22105, 35781, 57910, 93716, 151652, 245395, 397075, 642499, 1039604, 1682134, 2721770, 4403937, 7125742, 11529715, 18655494, 30185247, 48840780, 79026067, 127866888, 206892997

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) + 2 = 8;
a(3) = a(2) + a(1) + b(1) + 1 = 17.
Complement: (b(n)) = (2,4,5,6,7,9,10,11,12,13,14,15,16,18,...)

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] + 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A293316 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A001622 (golden ratio), A293076.

Sequence in context: A011850 A141422 A076980 * A294417 A001580 A360848

Adjacent sequences: A293054 A293055 A293056 * A293058 A293059 A293060

Keyword

nonn,easy

Author

Clark Kimberling, Oct 28 2017

Status

approved