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A295755 Solution of the complementary equation a(n) = a(n-1) + a(n-3) + a(n-4) + b(n-3), 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. 2
1, 2, 3, 4, 13, 25, 40, 66, 114, 190, 308, 502, 821, 1335, 2162, 3503, 5678, 9195, 14881, 24084, 38980, 63080, 102071, 165162, 267250, 432430, 699693, 1132136, 1831848, 2964004, 4795867, 7759886, 12555774, 20315682, 32871473, 53187172, 86058669, 139245866 (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.

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth-rate of the Fibonacci numbers (A000045).

LINKS

Table of n, a(n) for n=0..37.

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) + a(1) + 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] + a[n - 3] + 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}]   (* A295755 *)

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

CROSSREFS

Cf. A001622, A000045, A293411, A295754.

Sequence in context: A162222 A010346 A295621 * A089142 A123215 A051855

Adjacent sequences:  A295752 A295753 A295754 * A295756 A295757 A295758

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 30 2017

STATUS

approved

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Last modified May 30 18:07 EDT 2020. Contains 334728 sequences. (Running on oeis4.)