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A295360 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 3*b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 3, 5, 12, 18, 27, 41, 63, 98, 155, 247, 392, 628, 1008, 1624, 2620, 4228, 6831, 11041, 17853, 28874, 46706, 75559, 122244, 197778, 319996, 517747, 837715, 1355433, 2193118, 3548520, 5741606, 9290093, 15031665, 24321723 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..34.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

FORMULA

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

EXAMPLE

a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that

b(3) = 7 (least "new number")

a(3) = a(1) + a(0) + b(2) + b(1) - 3* b(0) = 12

Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)

MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; b[2] = 6;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - 3*b[n - 3];

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

z = 32; u = Table[a[n], {n, 0, z}]   (* A295360 *)

v = Table[b[n], {n, 0, 10}]  (* complement *)

CROSSREFS

Cf. A001622, A295357.

Sequence in context: A305552 A226652 A024696 * A197988 A025088 A279784

Adjacent sequences:  A295357 A295358 A295359 * A295361 A295362 A295363

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 21 2017

STATUS

approved

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Last modified July 26 16:44 EDT 2021. Contains 346294 sequences. (Running on oeis4.)