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%I #27 Feb 12 2021 16:03:29
%S 2,7,10,14,18,23,26,31,34,38,43,46,50,55,59,62,66,71,74,79,82,86,90,
%T 95,98,103,106,110,115,118,122,126,131,134,139,142,146,151,154,158,
%U 162,167,170,174,179,182,187,191,194,199,203,206,210,215,218,223,226
%N Solution (a(n)) of the complementary equation a(n) = b(n) + b(2n); see Comments.
%C 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,
%C (1) {a(n) - 4*n: n >= 0} = {2,3} and {3*b(n) - 4*n: n >= 0} = {2,3,4,5}.
%C (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}.
%C (3) {a(n) - a(n-1): n >= 1) = {h+k, h+k+1, h+k+2}.
%C (4) {k*b(n)-b(k*n): n >= 0} = {k-2, k-1, ..., 2*k-2}
%C ***
%C Guide to related sequences:
%C h k (a(n)) (b(n))
%C 1 2 A304799 A304800
%C 1 3 A304801 A304802
%C 1 4 A304803 A304804
%C 1 5 A304805 A304806
%C 2 3 A304807 A304808
%C 2 4 A304809 A304810
%C 2 5 A304811 A304812
%C 3 5 A304813 A304814
%C 4 5 A304815 A304816
%H Clark Kimberling, <a href="/A304799/b304799.txt">Table of n, a(n) for n = 0..10000</a>
%H C. Kimberling and P. J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Kimberling/kimb23.html">Complementary Equations with Advanced Subscripts</a>, J. Int. Seq. 24 (2021) Article 21.3.3.
%e 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.
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t h = 1; k = 2; a = {}; b = {1};
%t AppendTo[a, mex[Flatten[{a, b}], 1]];
%t Do[Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {k}];
%t AppendTo[a, Last[b] + b[[1 + (Length[b] - 1)/k h]]], {500}];
%t Take[a, 200] (* A304799 *)
%t Take[b, 200] (* A304800 *)
%t (* _Peter J. C. Moses_, May 14 2008 *)
%Y Cf. A304800-A304816.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, May 19 2018
%E Comments edited by _Clark Kimberling_, Jul 07 2019
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