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A304799 Solution (a(n)) of the complementary equation a(n) = b(n) + b(2n); see Comments. 19
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 (list; graph; refs; listen; history; text; internal format)
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

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 * A022886 A047524 A190447

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

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Last modified July 23 16:14 EDT 2019. Contains 325258 sequences. (Running on oeis4.)