login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 22 16:17 EDT 2019. Contains 326178 sequences. (Running on oeis4.)