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!)
A295615 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-1), where a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
2, 4, 6, 15, 33, 68, 130, 237, 417, 716, 1208, 2013, 3326, 5461, 8927, 14547, 23653, 38400, 62275, 100920, 163464, 264678, 428462, 693487, 1122324, 1816215, 2938973, 4755653, 7695123, 12451307, 20146996, 32598905, 52746540, 85346122, 138093378, 223440256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295613 for a guide to related sequences.

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

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

a(3) = 2*a(2) - a(0) + b(2) = 15

Complement: (b(n)) = (1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...)

MATHEMATICA

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

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

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}]  (* A295615 *)

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

CROSSREFS

Cf. A001622, A000045, A295613.

Sequence in context: A192536 A227902 A217525 * A127679 A049022 A209867

Adjacent sequences:  A295612 A295613 A295614 * A295616 A295617 A295618

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 May 10 22:15 EDT 2021. Contains 343780 sequences. (Running on oeis4.)