A304799 Solution (a(n)) of the complementary equation a(n) = b(n) + b(2n); see Comments.

2, 7, 10, 14, 18, 23, 26, 31, 34, 38, 43, 46, 50, 55, 59, 62, 66, 71, 74, 79, 82, 86, 90, 95, 98, 103, 106, 110, 115, 118, 122, 126, 131, 134, 139, 142, 146, 151, 154, 158, 162, 167, 170, 174, 179, 182, 187, 191, 194, 199, 203, 206, 210, 215, 218, 223, 226

Offset

0,1

Comments

Define complementary sequences a(n) and b(n) recursively: both are strictly increasing, b(0) = 1, and a(n) = b(n) + b(2n) for n >= 0. Empirically,

(1) {a(n) - 4*n: n >= 0} = {2,3} and {3*b(n) - 4*n: n >= 0} = {2,3,4,5}.

(2) If the equation for a(n) is generalized to a(n) = b(h*n) + b(k*n), where 1 <= h < k, then {a(n) - (h + k + 1)*n: n >= 0} = {2,3} and {(h + k)*b(n) - (h + k + 1)*n : n >= 0} = {k + h - 1, k + h, ..., 2*k + 2*h - 1}.

(3) {a(n) - a(n-1): n >= 1) = {h+k, h+k+1, h+k+2}.

(4) {k*b(n)-b(k*n): n >= 0} = {k-2, k-1, ..., 2*k-2}

***

Guide to related sequences:

h k (a(n)) (b(n))

1 2 A304799 A304800

1 3 A304801 A304802

1 4 A304803 A304804

1 5 A304805 A304806

2 3 A304807 A304808

2 4 A304809 A304810

2 5 A304811 A304812

3 5 A304813 A304814

4 5 A304815 A304816

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

C. Kimberling and P. J. C. Moses, Complementary Equations with Advanced Subscripts, J. Int. Seq. 24 (2021) Article 21.3.3.

Example

b(0) = 1, so that a(0) = 2.  Since a(1) = b(1) + b(2), we must have a(1) >= 7, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, and a(1) = 7.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
h = 1; k = 2; a = {}; b = {1};
AppendTo[a, mex[Flatten[{a, b}], 1]];
Do[Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {k}];
  AppendTo[a, Last[b] + b[[1 + (Length[b] - 1)/k h]]], {500}];
Take[a, 200]  (* A304799 *)
Take[b, 200]  (* A304800 *)
(* Peter J. C. Moses, May 14 2008 *)

Crossrefs

Cf. A304800-A304816.

Sequence in context: A130336 A303741 A085303 * A349499 A022886 A047524

Adjacent sequences: A304796 A304797 A304798 * A304800 A304801 A304802

Keyword

nonn,easy

Author

Clark Kimberling, May 19 2018

Extensions

Comments edited by Clark Kimberling, Jul 07 2019

Status

approved