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A293058 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 1
1, 3, 9, 19, 36, 64, 110, 185, 308, 507, 830, 1353, 2200, 3571, 5790, 9381, 15192, 24596, 39812, 64433, 104271, 168731, 273030, 441790, 714850, 1156671, 1871553, 3028257, 4899844, 7928136, 12828016, 20756189, 33584243, 54340472, 87924756, 142265270 (list; graph; refs; listen; history; text; internal format)
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) + 1 = 19.

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] + 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}]  (* A293316 *)

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

CROSSREFS

Cf. A001622 (golden ratio), A293076.

Sequence in context: A147174 A147158 A014540 * A294367 A146694 A146050

Adjacent sequences:  A293055 A293056 A293057 * A293059 A293060 A293061

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Oct 28 2017

STATUS

approved

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Last modified September 17 06:41 EDT 2019. Contains 327119 sequences. (Running on oeis4.)