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A295621 Solution of the complementary equation a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4) + b(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences. 3
1, 2, 3, 4, 13, 22, 55, 96, 201, 346, 659, 1117, 2015, 3372, 5882, 9752, 16643, 27411, 46093, 75559, 125754, 205448, 339432, 553177, 909097, 1478897, 2421000, 3933174, 6420218, 10419979, 16972319, 27525507, 44762106, 72554068, 117844772, 190931789, 309833797 (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.

LINKS

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

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

EXAMPLE

a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that

b(4) = 9 (least "new number")

a(4) = a(3) + 3*a(2) -2*a(1) - 2*a(0) + b(1) = 13

Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 14, 15, ...)

MATHEMATICA

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

a[0] = 1; a[1] = 2; a[2] = 3; a[3] = 4;

b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8;

a[n_] := a[n] = a[n - 1] + 3*a[n - 2] - 2*a[n - 3] - 2 a[n - 4] + b[n - 3];

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

z = 36;  Table[a[n], {n, 0, z}]   (* A295621 *)

Table[b[n], {n, 0, 20}]  (*complement *)

CROSSREFS

Cf. A001622, A000045, A295619, A295620.

Sequence in context: A236440 A162222 A010346 * A295755 A089142 A123215

Adjacent sequences:  A295618 A295619 A295620 * A295622 A295623 A295624

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 25 2017

STATUS

approved

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Last modified May 31 15:27 EDT 2020. Contains 334748 sequences. (Running on oeis4.)