login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A295613 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-1), where a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 6
1, 2, 3, 11, 27, 59, 116, 215, 383, 663, 1125, 1882, 3117, 5126, 8388, 13678, 22250, 36133, 58610, 94993, 153877, 249169, 403371, 652893, 1056646, 1709951, 2767040, 4477466, 7245014, 11723022, 18968613, 30692248, 49661511, 80354447, 130016685, 210371899 (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. Guide to related sequences:

A295613: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.

A295614: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.

A295615: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.

A295616: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.

A295617: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.

A295618: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.

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..35.

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

EXAMPLE

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

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

a(3) = 2*a(2) - a(0) + b(2) = 11

Complement: (b(n)) = (4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, ...)

MATHEMATICA

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

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

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

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

Table[a[n], {n, 0, 30}]   (* A295613 *)

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

CROSSREFS

Cf. A001622, A000045.

Sequence in context: A259428 A002981 A294637 * A232212 A232219 A265095

Adjacent sequences:  A295610 A295611 A295612 * A295614 A295615 A295616

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 25 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 April 15 08:18 EDT 2021. Contains 342977 sequences. (Running on oeis4.)