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A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 52
1, 3, 6, 13, 24, 44, 76, 129, 215, 355, 582, 951, 1548, 2515, 4080, 6613, 10712, 17345, 28078, 45445, 73546, 119016, 192588, 311631, 504247, 815907, 1320184, 2136122, 3456338, 5592493, 9048865, 14641393, 23690294, 38331724, 62022056, 100353819, 162375915 (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, which for each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

A293076:  a(n) = a(n-1) + a(n-2) + b(n-2)

A293316:  a(n) = a(n-1) + a(n-2) + b(n-2)

A293057:  a(n) = a(n-1) + a(n-2) + b(n-2)

A293058:  a(n) = a(n-1) + a(n-2) + b(n-2)

A293317:  a(n) = a(n-1) + a(n-2) + b(n-2)

A293349:  a(n) = a(n-1) + a(n-2) + b(n-2) + n

A293350:  a(n) = a(n-1) + a(n-2) + b(n-2) + 2n

A293351:  a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1

A293357:  a(n) = a(n-1) + a(n-2) + b(n-2) + n + 1

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

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

a(2) = a(1) + a(0) + b(0) = 3 + 1 + 2 = 6;

a(3) = a(2) + a(1) + b(1) = 6 + 3 + 4 = 13.

Complement: (b(n)) = (2,4,5,7,8,9,10,11,12,14,...)

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];

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

Table[a[n], {n, 0, 40}]  (* A293076 *)

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

CROSSREFS

Cf. A001622 (golden ratio), A293358.

Sequence in context: A120006 A263847 A061567 * A293421 A018081 A001452

Adjacent sequences:  A293073 A293074 A293075 * A293077 A293078 A293079

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Oct 28 2017

STATUS

approved

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Last modified July 3 20:41 EDT 2020. Contains 335418 sequences. (Running on oeis4.)